Weighted Extremal Domains and Best Rational Approximation
WWEIGHTED EXTREMAL DOMAINS AND BEST RATIONALAPPROXIMATION
LAURENT BARATCHART, HERBERT STAHL, AND MAXIM YATTSELEV
Abstract.
Let f be holomorphically continuable over the complex plane except forfinitely many branch points contained in the unit disk. We prove that best rationalapproximants to f of degree n , in the L -sense on the unit circle, have poles that asymp-totically distribute according to the equilibrium measure on the compact set outside ofwhich f is single-valued and which has minimal Green capacity in the disk among allsuch sets. This provides us with n -th root asymptotics of the approximation error. Byconformal mapping, we deduce further estimates in approximation by rational or mero-morphic functions to f in the L -sense on more general Jordan curves encompassing thebranch points. The key to these approximation-theoretic results is a characterization ofextremal domains of holomorphy for f in the sense of a weighted logarithmic potential,which is the technical core of the paper. List of Symbols
Sets : C extended complex plane T Jordan curve with exterior domain O and interior domain G T unit circle with exterior domain O and interior domain D E f set of the branch points of fK ∗ reflected set { z : 1 / ¯ z ∈ K } K set of minimal condenser capacity in K f ( G )Γ ν and D ν minimal set for Problem ( f, ν ) and its complement in C (Γ) (cid:15) { z ∈ D : dist( z, Γ) < (cid:15) } γ u image of a set γ under 1 / ( · − u ) Collections : K f ( G ) admissible sets for f ∈ A ( G ), K f = K f ( D ) G f admissible sets in K f comprised of a finite number of continuaΛ( F ) probability measures on F Mathematics Subject Classification.
Key words and phrases. rational approximation, meromorphic approximation, extremal domains, weakasymptotics, non-Hermitian orthogonality.The research of first and the third authors was partially supported by the ANR project “AHPI” (ANR-07-BLAN-0247-01). The research of the second author has been supported by the Deutsche Forschungsge-meinschaft (AZ: STA 299/13-1). a r X i v : . [ m a t h . C A ] A ug L. BARATCHART, H. STAHL, AND M. YATTSELEV
Spaces : P n algebraic polynomials of degree at most n M n ( G ) monic algebraic polynomials of degree n with n zeros in G , M n = M n ( D ) R n ( G ) R n ( G ) := P n − M − n ( G ), R n = R n ( D ) A ( G ) holomorphic functions C except for branch-type singularities in GL p ( T ) classical L p spaces, p < ∞ , with respect to arclength on T and the norm (cid:107) · (cid:107) p,T (cid:107) · (cid:107) K supremum norm on a set KE ( G ) Smirnov class of holomorphic functions in G with L traces on TE n ( G ) E n ( G ) := E ( G ) M − n ( G ) H classical Hardy space of holomorphic functions in D with L traces on T H n H n := H M − n Measures : ω ∗ reflected measure, ω ∗ ( B ) = ω ( B ∗ ) (cid:98) ω or (cid:101) ω balayage of ω , supp( ω ) ⊂ D , onto ∂Dω F equilibrium distribution on Fω F,ψ weighted equilibrium distribution on F in the field ψω ( F,E ) Green equilibrium distribution on F relative to C \ E Capacities :cap( K ) logarithmic capacity of K cap ν ( K ) ν -capacity of K cap( E, F ) capacity of the condenser (
E, F ) Energies : I [ ω ] logarithmic energy of ωI ψ [ ω ] weighted logarithmic energy of ω in the field ψI D [ ω ] Green energy of ω relative to DI ν [ K ] ν -energy of a set K D D ( u, v ) Dirichlet integral of functions u, v in a domain D Potentials : V ω logarithmic potential of ωV ω ∗ spherical logarithmic potential of ωU ν spherically normalized logarithmic potential of ν ∗ V ωD Green potential of ω relative to Dg D ( · , u ) Green function for D with pole at u Constants : c ( ψ ; F ) modified Robin constant, c ( ψ ; F ) = I ψ [ ω F,ψ ] − (cid:82) ψdω F,ψ c ( ν ; D ) is equal to (cid:82) g D ( z, ∞ ) dν ( z ) if D is unbounded and to 0 otherwise1. Introduction
Approximation theory in the complex domain has undergone striking developments overthe last years that gave new impetus to this classical subject. After the solution to theGonchar conjecture [39, 44] and the achievement of weak asymptotics in Pad´e approxima-tion [48, 50, 25] came the disproof of the Baker-Gammel-Wills conjecture [36, 15], and theRiemann-Hilbert approach to the limiting behavior of orthogonal polynomials [18, 31] thatopened the way to unprecedented strong asymptotics in rational interpolation [4, 3, 14] (see[17, 30] for other applications of this powerful device). Meanwhile, the spectral approach tomeromorphic approximation [1], already instrumental in [39], has produced sharp conversetheorems in rational approximation and fueled engineering applications to control systemsand signal processing [23, 41, 38, 40].
ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 3
In most investigations involved with non-Hermitian orthogonal polynomials and rationalinterpolation, a central role has been played by certain geometric extremal problems fromlogarithmic potential theory, close in spirit to the Lavrentiev type [32], that were introducedin [49]. On the one hand, their solution produces systems of arcs over which non-Hermitianorthogonal polynomials can be analyzed; on the other hand such polynomials are preciselydenominators of rational interpolants to functions that may be expressed as Cauchy integralsover this system of arcs, the interpolation points being chosen in close relation with the latter.One issue facing now the theory is to extend to best rational or meromorphic approximantsof prescribed degree to a given function the knowledge that was gained about rationalinterpolants. Optimality may of course be given various meanings. However, in view ofthe connections with interpolation theory pointed out in [35, 11, 12], and granted theirrelevance to spectral theory, the modeling of signals and systems, as well as inverse problems[2, 22, 28, 37, 10, 29, 46], it is natural to consider foremost best approximants in Hardyclasses.The main interest there attaches to the behavior of the poles whose determination is thenon-convex and most difficult part of the problem. The first obstacle to value interpolationtheory in this context is that it is unclear whether best approximants of a given degreeshould interpolate the function at enough points, and even if they do these interpolationpoints are no longer parameters to be chosen adequately in order to produce convergence butrather unknown quantities implicitly determined by the optimality property. The presentpaper deals with H -best rational approximation in the complement of the unit disk, forwhich maximum interpolation is known to take place; it thus remains in this case to locatethe interpolation points. This we do asymptotically, when the degree of the approximantgoes large, for functions whose singularities consist of finitely many poles and branch pointsin the disk. More precisely, we prove that the normalized (probability) counting measuresof the poles of the approximants converge, in the weak star sense, to the equilibrium dis-tribution of the continuum of minimum Green capacity, in the disk, outside of which theapproximated function is single-valued. By conformal mapping, the result carries over tobest meromorphic approximants with a prescribed number of poles, in the L -sense on a Jor-dan curve encompassing the poles and branch points. We also estimate the approximationerror in the n -th root sense, that turns out to be the same as in uniform approximation forthe functions under consideration. Note that H -best rational approximants on the disk areof fundamental importance in stochastic identification [28] and that functions with branchpoints arise naturally in inverse sources and potential problems [7, 9], so the result may beregarded as a prototypical case of the above-mentioned program.The paper is organized as follows. In Sections 2 and 3, we fix the terminology andrecall some known facts about H -best rational approximants and sets of minimal condensercapacity, before stating our main results (Theorems 5 and 7) along with some corollaries.We set up in Section 4 a weighted version of the extremal potential problem introduced in[49] ( cf. Definition 9) and stress its main features. Namely, a solution exists uniquely andcan be characterized, among continua outside of which the approximated function is single-valued, as a system of arcs possessing the so-called S -property in the field generated by theweight ( cf. Definition 10 and Theorem 12). Section 5 is a brief introduction to multipointPad´e interpolants, of which H -best rational approximants are a particular case. Section 6contains the proofs of all the results: first we establish Theorem 12, which is the technicalcore of the paper, using compactness properties of the Hausdorff metric together with the a priori geometric estimate of Lemma 17 to prove existence; the S -property is obtainedby showing the local equivalence of our weighted extremal problem with one of minimal L. BARATCHART, H. STAHL, AND M. YATTSELEV condenser capacity (Lemma 19); uniqueness then follows from a variational argument usingDirichlet integrals (Lemma 20). After Theorem 12 is established, the proof of Theorem 7 isnot too difficult. We choose as weight (minus) the potential of a limit point of the normalizedcounting measures of the interpolation points of the approximants and, since we now knowthat a compact set of minimal weighted capacity exists and that it possesses the S -property,we can adapt results from [25] to the effect that the normalized counting measures of thepoles of the approximants converge to the weighted equilibrium distribution on this systemof arcs. To see that this is nothing but the Green equilibrium distribution, we appeal to thefact that poles and interpolation points are reflected from each other across the unit circlein H -best rational approximation. The results carry over to more general domains as inTheorem 5 by a conformal mapping (Theorem 6). The appendix in Section 7 gathers sometechnical results from logarithmic potential theory that are needed throughout the paper.2. Rational Approximation in L In this work we are concerned with rational approximation of functions analytic at in-finity having multi-valued meromorphic continuation to the entire complex plane deprivedof a finite number of points. The approximation will be understood in the L -norm on arectifiable Jordan curve encompassing all the singularities of the approximated function.Namely, let T be such a curve. Let further G and O be the interior and exterior domains of T , respectively, i.e., the bounded and unbounded components of the complement of T in theextended complex plane C . We denote by L ( T ) the space of square-summable functions on T endowed with the usual norm (cid:107) f (cid:107) ,T := (cid:90) T | f | ds, where ds is the arclength differential. Set P n to be the space of algebraic polynomials ofdegree at most n and M n ( G ) to be its subset consisting of monic polynomials with n zerosin G . Define(2.1) R n ( G ) := (cid:26) p ( z ) q ( z ) = p n − z n − + p n − z n − + · · · + p z n + q n − z n − + · · · + q : p ∈ P n − , q ∈ M n ( G ) (cid:27) . That is, R n ( G ) is the set of rational functions with at most n poles that are holomorphicin some neighborhood of O and vanish at infinity. Let f be a function holomorphic andvanishing at infinity (vanishing at infinity is a normalization required for convenience only).We say that f belongs to the class A ( G ) if(i) f admits holomorphic and single-valued continuation from infinity to an open neigh-borhood of O ; (ii) f admits meromorphic, possibly multi-valued, continuation along any arc in G \ E f starting from T , where E f is a finite set of points in G ; (iii) E f is non-empty, the meromorphic continuation of f from infinity has a branchpoint at each element of E f . The primary example of functions in A ( G ) is that of algebraic functions. Every algebraicfunction f naturally defines a Riemann surface. Fixing a branch of f at infinity is equivalentto selecting a sheet of this covering surface. If all the branch points and poles of f on thissheet lie above G , the function f belongs to A ( G ). Other functions in A ( G ) are those ofthe form g ◦ log( l /l ) + r , where g is entire and l , l ∈ M m ( G ) while r ∈ R k ( G ) for some m, k ∈ N . However, A ( G ) is defined in such a way that it contains no function in R n ( G ), n ∈ N , in order to avoid degenerate cases. ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 5
With the above notation, the goal of this section is to describe the asymptotic behaviorof(2.2) ρ n, ( f, T ) := inf {(cid:107) f − r (cid:107) ,T : r ∈ R n ( G ) } , f ∈ A ( G ) . This problem is, in fact, a variation of a classical question in Chebyshev (uniform) rationalapproximation of holomorphic functions where it is required to describe the asymptoticbehavior of ρ n, ∞ ( f, T ) := inf {(cid:107) f − r (cid:107) T : r ∈ R n ( G ) } , where (cid:107) · (cid:107) T is the supremum norm on T . The theory behind Chebyshev approximation israther well established while its L -counterpart, which naturally arises in system identifi-cation and control theory [5] and serves as a method to approach inverse source problems[7, 9, 10], is not so much developed. In particular, it follows from the techniques of rationalinterpolation devised by Walsh [54] that(2.3) lim sup n →∞ ρ /nn, ∞ ( f, T ) ≤ exp (cid:26) − K, T ) (cid:27) for any function f holomorphic outside of K ⊂ G , where cap( K, T ) is the condenser capacity(Section 7.1.3) of a set K contained in a domain G relative to this domain . On the otherhand, it was conjectured by Gonchar and proved by Parf¨enov [39, Sec. 5] on simply connecteddomains, also later by Prokhorov [44] in full generality, that(2.4) lim inf n →∞ ρ / nn, ∞ ( f, T ) ≤ exp (cid:26) − K, T ) (cid:27) . Notice that only the n -th root is taken in (2.3) while (2.4) provides asymptotics for the2 n -th root. Observe also that there are many compacts K which make a given f ∈ A ( G )single-valued in their complement. Hence, (2.3) and (2.4) can be sharpened by taking theinfimum over K on the right-hand side of both inequalities. To explore this fact we needthe following definition. Definition 1.
We say that a compact K ⊂ G is admissible for f ∈ A ( G ) if C \ K isconnected and f has meromorphic and single-valued extension there. The collection of alladmissible sets for f we denote by K f ( G ) . As equations (2.3) and (2.4) suggest and Theorem 5 below shows, the relevant admissibleset in rational approximation to f ∈ A ( G ) is the set of minimal condenser capacity [48, 49,50, 51] relative to G : Definition 2.
Let f ∈ A ( G ) . A compact K ∈ K f ( G ) is said to be a set of minimal condensercapacity for f if (i) cap(K , T ) ≤ cap( K, T ) for any K ∈ K f ( G ) ; (ii) K ⊂ K for any K ∈ K f ( G ) such that cap( K, T ) = cap(K , T ) . It follows from the properties of condenser capacity that cap(K , T ) = cap( T, K) =cap( O, K) since K has connected complement that contains T by Definition 1. In otherwords, the set K can be seen as the complement of the “largest” (in terms of capacity)domain containing O on which f is single-valued and meromorphic. In fact, this is exactlythe point of view taken up in [48, 49, 50, 51]. It is known that such a set always exists, is In Section 7 the authors provide a concise but self-contained account of logarithmic potential theory.The reader may want to consult this section to get accustomed with the employed notation for capacities,energies, potentials, and equilibrium measures.
L. BARATCHART, H. STAHL, AND M. YATTSELEV unique, and has, in fact, a rather special structure. To describe it, we need the followingdefinition.
Definition 3.
We say that a set K ∈ K f ( G ) is a smooth cut for f if K = E ∪ E ∪ (cid:83) γ j ,where (cid:83) γ j is a finite union of open analytic arcs, E ⊆ E f and each point in E is theendpoint of exactly one γ j , while E is a finite set of points each element of which is theendpoint of at least three arcs γ j . Moreover, we assume that across each arc γ j the jump of f is not identically zero. Let us informally explain the motivation behind Definition 3. In order to make f single-valued, it is intuitively clear that one needs to choose a proper system of cuts joiningcertain points in E f so that one cannot encircle these points nor access the remainingones without crossing the cut. It is then plausible that the geometrically “smallest” systemof cuts comprises of Jordan arcs. In the latter situation, the set E consists of the points ofintersection of these arcs. Thus, each element of E serves as an endpoint for at least threearcs since two arcs meeting at a point are considered to be one. In Definition 3 we alsoimpose that the arcs be analytic. It turns out that the set of minimal condenser capacity(Theorem S) as well as minimal sets from Section 4 (Theorem 12) have exactly this structure.It is possible for E to be a proper subset of E f . This can happen when some of the branchpoints of f lie above G but on different sheets of the Riemann surface associated with f that cannot be accessed without crossing the considered system of cuts.The following is known about the set K (Definition 2) [48, Thm. 1 and 2] and [49, Thm. 1]. Theorem S.
Let f ∈ A ( G ) . Then K , the set of minimal condenser capacity for f , existsand is unique. Moreover, it is a smooth cut for f and (2.5) ∂∂ n + V ω ( T, K) C \ K = ∂∂ n − V ω ( T, K) C \ K on (cid:91) γ j , where ∂/∂ n ± are the partial derivatives with respect to the one-sided normals on each γ j , V ω ( T, K) C \ K is the Green potential of ω ( T, K) relative to C \ K , and ω ( T, K) is the Green equilibriumdistribution on T relative to C \ K (Section 7.1.3). Note that (2.5) is independent of the orientation chosen on γ j to define ∂/∂ n ± . Property(2.5) turns out to be more beneficial than Definition 2 in the sense that all the forthcomingproofs use only (2.5). However, one does not achieve greater generality by relinquishingthe connection to the condenser capacity and considering (2.5) by itself as this propertyuniquely characterizes K. Indeed, the following theorem is proved in Section 6.4. Theorem 4.
The set of minimal condenser capacity for f ∈ A ( G ) is uniquely characterizedas a smooth cut for f that satisfies (2.5) . With all the necessary definitions at hand, the following result takes place.
Theorem 5.
Let T be a rectifiable Jordan curve with interior domain G and exterior domain O . If f ∈ A ( G ) , then (2.6) lim n →∞ ρ / nn, ( f, T ) = lim n →∞ ρ / nn, ∞ ( f, T ) = exp (cid:26) − , T ) (cid:27) , where K is set of minimal condenser capacity for f . Since the arcs γ j are analytic and the potential V ω ( T, K) C \ K is identically zero on them, V ω ( T, K) C \ K can beharmonically continued across each γ j by reflection. Hence, the partial derivatives in (2.5) exist and arecontinuous. ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 7
The second equality in (2.6) follows from [25, Thm 1 (cid:48) ], where a larger class of functionsthan A ( G ) is considered (see Theorem GR in Section 6.3). To prove the first equality, weappeal to another type of approximation, namely, meromorphic approximation in L -normon T , for which asymptotics of the error and the poles are obtained below. This type ofapproximation turns out to be useful in certain inverse source problems [9, 34, 16]. Observethat | T | /p − / (cid:107) h (cid:107) ,T ≤ (cid:107) h (cid:107) p,T ≤ | T | /p (cid:107) h (cid:107) T for any p ∈ (2 , ∞ ) and any bounded function h on T by H¨older inequality, where (cid:107) · (cid:107) p,T is the usual p -norm on T with respect to ds and | T | is the arclength of T . Thus, Theorem 5 implies that (2.6) holds for L p ( T )-best rationalapproximants as well when p ∈ (2 , ∞ ). In fact, as Vilmos Totik pointed out to the authors[53], with a different method of proof Theorem 5 can be extended to include the full range p ∈ [1 , ∞ ].Just mentioned best meromorphic approximants are defined as follows. Denote by E ( G )the Smirnov class for G [20, Sec. 10.1]. It is known that functions in E ( G ) have non-tangential boundary values a.e. on T and thus formed traces of functions in E ( G ) belongto L ( T ). Now, put E n ( G ) := E ( G ) M − n ( G ) to be the set of meromorphic functions in G with at most n poles there and square-summable traces on T . It is known [10, Sec. 5] thatfor each n ∈ N there exists g n ∈ E n ( G ) such that (cid:107) f − g n (cid:107) ,T = inf (cid:8) (cid:107) f − g (cid:107) ,T : g ∈ E n ( G ) (cid:9) . That is, g n is a best meromorphic approximant for f in the L -norm on T . Theorem 6.
Let T be a rectifiable Jordan curve with interior domain G and exterior domain O . If f ∈ A ( G ) , then (2.7) | f − g n | / n cap → exp (cid:26) V ω (K ,T ) G − , T ) (cid:27) in G \ K , where the functions g n ∈ E n ( G ) are best meromorphic approximants to f in the L -norm on T , K is the set of minimal condenser capacity for f in G , ω (K ,T ) is the Green equilibriumdistribution on K relative to G , and cap → denotes convergence in capacity (see Section 7.1.1).Moreover, the counting measures of the poles of g n converge weak ∗ to ω (K ,T ) .
3. ¯ H -Rational Approximation To prove Theorems 5 and 6, we derive a stronger result in the model case where G is theunit disk, D . The strengthening comes from the facts that in this case L -best meromorphicapproximants specialize to L -best rational approximants the latter also turn out to beinterpolants. In fact, we consider not only best rational approximants but also criticalpoints in rational approximation.Let T be the unit circle and set for brevity L := L ( T ). Denote by H ⊂ L the Hardyspace of functions whose Fourier coefficients with strictly negative indices are zero. Thespace H can be described as the set of traces of holomorphic functions in the unit diskwhose square-means on concentric circles centered at zero are uniformly bounded above [20]. Further, denote by ¯ H the orthogonal complement of H in L , L = H ⊕ ¯ H , withrespect to the standard scalar product (cid:104) f, g (cid:105) := (cid:90) T f ( τ ) g ( τ ) | dτ | , f, g ∈ L . A function h belongs to E ( G ) if h is holomorphic in G and there exists a sequence of rectifiable Jordancurves, say { T n } , whose interior domains exhaust G , such that (cid:107) h (cid:107) ,T n ≤ const. independently of n . Each such function has non-tangential boundary values almost everywhere on T and can be recoveredfrom these boundary values by means of the Cauchy or Poisson integral. L. BARATCHART, H. STAHL, AND M. YATTSELEV
From the viewpoint of analytic function theory, ¯ H can be regarded as a space of tracesof functions holomorphic in O := C \ D and vanishing at infinity whose square-means onthe concentric circles centered at zero (this time with radii greater then 1) are uniformlybounded above. In what follows, we denote by (cid:107) · (cid:107) the norm on L induced by the scalarproduct (cid:104)· , ·(cid:105) . In fact, (cid:107) · (cid:107) is a norm on H and ¯ H as well.We set M n := M n ( D ) and R n := R n ( D ). Observe that R n is the set of rational functionsof degree at most n belonging to ¯ H . With the above notation, consider the following ¯ H -rational approximation problem: Given f ∈ ¯ H and n ∈ N , minimize (cid:107) f − r n (cid:107) over all r ∈ R n . It is well-known (see [6, Prop. 3.1] for the proof and an extensive bibliography on thesubject) that this minimum is always attained while any minimizing rational function, alsocalled a best rational approximant to f , lies in R n \ R n − unless f ∈ R n − .Best rational approximants are part of the larger class of critical points in ¯ H -rationalapproximation. From the computational viewpoint, critical points are as important as bestapproximants since a numerical search is more likely to yield a locally best rather than abest approximant. For fixed f ∈ ¯ H , critical points can be defined as follows. Set(3.1) Ψ f,n : P n − × M n → [0 , ∞ )( p, q ) (cid:55)→ (cid:107) f − p/q (cid:107) . In other words, Ψ f,n is the squared error of approximation of f by r = p/q in R n . Wetopologically identify P n − × M n with an open subset of C n with coordinates p j and q k , j, k ∈ { , . . . , n − } (see (2.1)). Then a pair of polynomials ( p c , q c ) ∈ P n − × M n , identifiedwith a vector in C n , is said to be a critical pair of order n , if all the partial derivatives of Ψ f,n do vanish at ( p c , q c ). Respectively, a rational function r c ∈ R n is a critical point of order n ifit can be written as the ratio r c = p c /q c of a critical pair ( p c , q c ) in P n − × M n . A particularexample of a critical point is a locally best approximant . That is, a rational function r l = p l /q l associated with a pair ( p l , q l ) ∈ P n − × M n such that Ψ f,n ( p l , q l ) ≤ Ψ f,n ( p, q ) for all pairs( p, q ) in some neighborhood of ( p l , q l ) in P n − × M n . We call a critical point of order n irreducible if it belongs to R n \ R n − . As we have already mentioned, best approximants,as well as local minima, are always irreducible critical points unless f ∈ R n − . In generalthere may be other critical points, reducible or irreducible, which are saddles or maxima. Infact, to give amenable conditions for uniqueness of a critical point it is a fairly open problemof great practical importance, see [5, 11, 13] and the bibliography therein.One of the most important properties of critical points is the fact that they are “maximal”rational interpolants. More precisely, let f ∈ ¯ H and r n be an irreducible critical point oforder n , then r n interpolates f at the reflection ( z (cid:55)→ / ¯ z ) of each pole of r n with order twicethe multiplicity that pole [35], [13, Prop. 2], which is the maximal number of interpolationconditions ( i.e., n ) that can be imposed in general on a rational function of type ( n − , n )(i.e., the ratio of a polynomial of degree n − n ).With all the definitions at hand, we are ready to state our main results concerning thebehavior of critical points in ¯ H -rational approximation for functions in A ( D ), which willbe proven in Section 6.4. Theorem 7.
Let f ∈ A ( D ) and { r n } n ∈ N be a sequence of irreducible critical points in ¯ H -rational approximation for f . Further, let K be the set of minimal condenser capacity for f . Then the normalized counting measures of the poles of r n converge weak ∗ to the Green The normalized counting measure of poles/zeros of a given function is a probability measure havingequal point masses at each pole/zero of the function counting multiplicity.
ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 9 equilibrium distribution on K relative to D , ω (K , T ) . Moreover, it holds that (3.2) | ( f − r n ) | / n cap → exp (cid:110) − V ω ∗ (K , T ) C \ K (cid:111) in C \ (K ∪ K ∗ ) , where K ∗ and ω ∗ (K , T ) are the reflections of K and ω (K , T ) across T , respectively, and cap → denotes convergence in capacity. In addition, it holds that (3.3) lim sup n →∞ | ( f − r n )( z ) | / n ≤ exp (cid:110) − V ω ∗ (K , T ) C \ K ( z ) (cid:111) uniformly for z ∈ O . Using the fact that the Hardy space H is orthogonal to ¯ H , one can show that L -bestmeromorphic approximants discussed in Theorem 6 specialize to L -best rational approxi-mants when G = D (see the proof of Theorem 6). Moreover, it is shown in Lemma 25 inSection 7 that − V ω ∗ (K , T ) C \ K ≡ V ω (K , T ) D − / cap(K , T ) in D . So, formula (2.7) is, in fact, a gener-alization of (3.2), but only in G \ K. Lemma 25 also implies that V ω ∗ (K , T ) C \ K ≡ / cap(K , T ) on T . In particular, the following corollary to Theorem 7 can be stated. Corollary 8.
Let f , { r n } , and K be as in Theorem 7. Then (3.4) lim n →∞ (cid:107) f − r n (cid:107) / n = lim n →∞ (cid:107) f − r n (cid:107) / n T = exp (cid:26) − , T ) (cid:27) , where (cid:107) · (cid:107) T stands for the supremum norm on T . Observe that Corollary 8 strengthens Theorem 5 in the case when T = T . Indeed,(3.4) combined with (2.6) implies that the critical points in ¯ H -rational approximation alsoprovide the best rate of uniform approximation in the n -th root sense for f on O .4. Domains of Minimal Weighted Capacity
Our approach to Theorem 7 lies in exploiting the interpolation properties of the criticalpoints in ¯ H -rational approximation. To this end we first study the behavior of rationalinterpolants with predetermined interpolation points (Theorem 14 in Section 5). However,before we are able to touch upon the subject of rational interpolation proper, we need toidentify the corresponding minimal sets. These sets are the main object of investigation inthis section.Let ν be a probability Borel measure supported in D . We set(4.1) U ν ( z ) := − (cid:90) log | − z ¯ u | dν ( u ) . The function U ν is simply the spherically normalized logarithmic potential of ν ∗ , the re-flection of ν across T (see (7.1)). Hence, it is a harmonic function outside of supp( ν ∗ ), inparticular, in D . Considering − U ν as an external field acting on non-polar compact subsetsof D , we define the weighted capacity in the usual manner (Section 7.1.2). Namely, for sucha set K ⊂ D , we define the ν -capacity of K by(4.2) cap ν ( K ) := exp {− I ν [ K ] } , I ν [ K ] := min ω (cid:18) I [ ω ] − (cid:90) U ν dω (cid:19) , For every set K we define the reflected set K ∗ as K ∗ := { z : 1 / ¯ z ∈ K } . If ω is a Borel measure in C ,then ω ∗ is a measure such that ω ∗ ( B ) = ω ( B ∗ ) for every Borel set B . where the minimum is taken over all probability Borel measures ω supported on K (seeSection 7.1.1 for the definition of energy I [ · ]). Clearly, U δ ≡ δ ( · ) issimply the classical logarithmic capacity (Section 7.1.1), where δ is the Dirac delta at theorigin.The purpose of this section is to extend results in [48, 49] obtained for ν = δ . For that,we introduce a notion of a minimal set in a weighted context. This generalization is the keyenabling us to adapt the results of [25] to the present situation, and its study is really thetechnical core of the paper. For simplicity, we put K f := K f ( D ). Definition 9.
Let ν be a probability Borel measure supported in D . A compact Γ ν ∈ K f , f ∈ A ( D ) , is said to be a minimal set for Problem ( f, ν ) if (i) cap ν (Γ ν ) ≤ cap ν ( K ) for any K ∈ K f ; (ii) Γ ν ⊂ Γ for any Γ ∈ K f such that cap(Γ) = cap(Γ ν ) . The set Γ ν will turn out to have geometric properties similar to those of minimal condensercapacity sets (Definition 2). This motivates the following definition. Definition 10.
A compact Γ ∈ K f is said to be symmetric with respect to a Borel measure ω , supp( ω ) ∩ Γ = ∅ , if Γ is a smooth cut for f (Definition 3) and (4.3) ∂∂ n + V ω C \ Γ = ∂∂ n − V ω C \ Γ on (cid:91) γ j , where ∂/∂ n ± are the partial derivatives with respect to the one-sided normals on each sideof γ j and V ω C \ Γ is the Green potential of ω relative to C \ Γ . Definition 10 is given in the spirit of [49] and thus appears to be different from the
S-property defined in [25]. Namely, a compact Γ ⊂ D having the structure of a smooth cut issaid to possess the S-property in the field ψ , assumed to be harmonic in some neighborhoodof Γ, if(4.4) ∂ ( V ω Γ ,ψ + ψ ) ∂ n + = ∂ ( V ω Γ ,ψ + ψ ) ∂ n − , q. e. on (cid:91) γ j , where ω Γ ,ψ is the weighted equilibrium distribution on Γ in the field ψ and the normalderivatives exist at every tame point of supp( ω Γ ,ψ ) (see Section 6.3). It follows from (7.23)and (7.20) that Γ has the S-property in the field − U ν if and only if it is symmetric withrespect to ν ∗ , taking into account that V ω Γ , − Uν − U ν is constant on the arcs γ j which areregular (see Section 7.2.2) hence the normal derivatives exist at every point. This reconcilesDefinition 10 with the one given in [25] in the setting of our work.The symmetry property (4.3) entails that V ω C \ Γ has a very special structure. Proposition 11.
Let
Γ = E ∪ E ∪ (cid:83) γ j and V ω C \ Γ be as in Definitions 3 and 10. Thenthe arcs γ j possess definite tangents at their endpoints. The tangents to the arcs ending at e ∈ E (there are at least three by definition of a smooth cut) are equiangular. Further, set (4.5) H ω, Γ := ∂ z V ω C \ Γ , ∂ z := ( ∂ x − i∂ y ) / . Then H ω, Γ is holomorphic in C \ (Γ ∪ supp( ω )) and has continuous boundary values from eachside of every γ j that satisfy H + ω, Γ = − H − ω, Γ on each γ j . Moreover, H ω, Γ is a meromorphicfunction in C \ supp ( ω ) that has a simple pole at each element of E and a zero at eachelement e of E whose order is equal to the number of arcs γ j having e as endpoint minus2. ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 11
The following theorem is the main result of this section and is a weighted generalizationof [48, Thm. 1 and 2] and [49, Thm. 1] for functions in A ( D ). Theorem 12.
Let f ∈ A ( D ) and ν be a probability Borel measure supported in D . Thena minimal set for Problem ( f, ν ) , say Γ ν , exists, is unique and contained in D r , r :=max z ∈ E f | z | . Moreover, Γ ∈ K f is minimal if and only if it is symmetric with respectto ν ∗ . The proof of Theorem 12 is carried out in Section 6.1 and the proof of Proposition 11 ispresented in Section 6.2. 5.
Multipoint Pad´e Approximation
In this section, we state a result that yields complete information on the n -th root be-havior of rational interpolants to functions in A ( D ). It is essentially a consequence bothof Theorem 12 and Theorem 4 in [25] on the behavior of multipoint Pad´e approximants tofunctions analytic off a symmetric contour, whose proof plays here an essential role.Classically, diagonal multipoint Pad´e approximants to f are rational functions of type( n, n ) that interpolate f at a prescribed system of 2 n + 1 points. However, when theapproximated function is holomorphic at infinity, as is the case f ∈ A ( D ), it is customaryto place at least one interpolation point there. More precisely, let E = { E n } be a triangularscheme of points in C \ E f and let v n be the monic polynomial with zeros at the finite pointsof E n . In other words, E := { E n } n ∈ N is such that each E n consists of 2 n not necessarilydistinct nor finite points contained in C \ E f . Definition 13.
Given f ∈ A ( D ) and a triangular scheme E , the n -th diagonal Pad´e ap-proximant to f associated with E is the unique rational function Π n = p n /q n satisfying: • deg p n ≤ n , deg q n ≤ n , and q n (cid:54)≡ ; • ( q n ( z ) f ( z ) − p n ( z )) /v n ( z ) has analytic (multi-valued) extension to C \ E f ; • ( q n ( z ) f ( z ) − p n ( z )) /v n ( z ) = O (cid:0) /z n +1 (cid:1) as z → ∞ . Multipoint Pad´e approximants always exist since the conditions for p n and q n amount tosolving a system of 2 n + 1 homogeneous linear equations with 2 n + 2 unknown coefficients,no solution of which can be such that q n ≡ q n is monic); notethat the required interpolation at infinity is entailed by the last condition and therefore Π n is, in fact, of type ( n − , n ).We define the support of E as supp( E ) := ∩ n ∈ N ∪ k ≥ n E k . Clearly, supp( E ) contains thesupport of any weak ∗ limit point of the normalized counting measures of points in E n (seeSection 7.2.5). We say that a Borel measure ω is the asymptotic distribution for E if thenormalized counting measures of points in E n converge to ω in the weak ∗ sense. Theorem 14.
Let f ∈ A ( D ) and ν be a probability Borel measure supported in D . Further,let E be a triangular scheme of points, supp( E ) ⊂ O , with asymptotic distribution ν ∗ . Then (5.1) | f − Π n | / n cap → exp (cid:110) − V ν ∗ D ν (cid:111) in D ν \ supp( ν ∗ ) , D ν = C \ Γ ν , where Π n are the diagonal Pad´e approximants to f associated with E and Γ ν is the minimalset for Problem ( f, ν ) . It also holds that the normalized counting measures of poles of Π n converge weak ∗ to (cid:98) ν ∗ , the balayage (Section 7.2) of ν ∗ onto Γ ν relative to D ν . In particular,the poles of Π n tend to Γ ν in full proportion. Proofs
Proof of Theorem 12.
In this section we prove Theorem 12 in several steps that areorganized as separate lemmas.Denote by G f the subset of K f comprised of those admissible sets that are unions ofa finite number of disjoint continua each of which contains at least two point of E f . Inparticular, each member of G f is a regular set [45, Thm. 4.2.1] and cap(Γ \ (Γ ∩ Γ )) > (cid:54) = Γ , Γ , Γ ∈ G f (if Γ (cid:54) = Γ , there exists a continuum γ ⊂ Γ \ (Γ ∩ Γ ); as anycontinuum has positive capacity [45, Thm. 5.3.2], the claim follows). Considering G f insteadof K f makes the forthcoming analysis simpler but does not alter the original problem as thefollowing lemma shows. Lemma 15.
It holds that inf Γ ∈G f cap ν (Γ) = inf K ∈K f cap ν ( K ) .Proof. Pick K ∈ K f and let O be the collection of all domains containing C \ K to which f extends meromorphically. The set O is nonempty as it contains C \ K , it is partiallyordered by inclusion, and any totally ordered subset { O α } has an upper bound, e.g. ∪ α O α .Therefore, by Zorn’s lemma [33, App. 2, Cor.2.5], O has a maximal element, say O .Put F = C \ O . With a slight abuse of notation, we still denote by f the meromorphiccontinuation of the latter to C \ F . Note that a point in E f is either “inactive” (i.e., is nota branch point for that branch of f that we consider over C \ F ) or belongs to F .If F is not connected, there are two bounded disjoint open sets V , V such that ( V ∪ V ) ∩ F = F and, for j = 1 , ∂V j ∩ F = ∅ , V j ∩ F (cid:54) = ∅ . If V j contains only one connectedcomponent of F , we do not refine it further. Otherwise, there are two disjoint open sets V j, , V j, ⊂ V j such that ( V j, ∪ V j, ) ∩ F = V j ∩ F and, for k = 1 , ∂V j,k ∩ F = ∅ , V j,k ∩ F (cid:54) = ∅ . Iterating this process, we obtain successive generations of bounded finitedisjoint open covers of F , each element of which contains at least one connected componentof F and has boundary that does not meet F . The process stops if F has finitely manycomponents, and then the resulting open sets separate them. Otherwise the process cancontinue indefinitely and, if C , . . . , C N are the finitely many connected components of F that meet E f , at least one open set of the N + 1-st generation contains no C j . In any case,if F has more than N connected components, there is a bounded open set V , containing atleast one connected component of F and no point of E f ∩ F , such that ∂V ∩ F = ∅ .Let A be the unbounded connected component of C \ V and A , . . . , A L those boundedcomponents of C \ V , if any, that contain some C j (if L = 0 this is the empty collection).Since O = C \ F is connected, each ∂A (cid:96) can be connected to ∂A by a closed arc γ (cid:96) ⊂ O .Then W := V \∪ (cid:96) γ (cid:96) is open with ∂W ∩ F = ∅ , it contains at least one connected componentof F , and no bounded component of its complement meets E f ∩ F . Let X be the unboundedconnected component of C \ W and put U := C \ X . The set U is open, simply connected,and ∂U ⊂ ∂W is compact and does not meet F . Moreover, since it is equal to the union of W and all the bounded components of C \ W , U does not meet E f ∩ F .Now, f is defined and meromorphic in a neighborhood of ∂U ⊂ O , and meromorphicallycontinuable along any path in U since the latter contains no point of E f ∩ F . Since U issimply connected, f extends meromorphically to O ∪ U by the monodromy theorem. Howeverthe latter set is a domain which strictly contains O since U contains W and thus at least oneconnected component of F . This contradicts the maximality of O and shows that F consistsprecisely of N connected components, namely C , . . . , C N . Moreover, if Γ j is a Jordan curveencompassing C j and no other C (cid:96) , then by what precedes f must be single-valued along Γ j which is impossible if C j ∩ E f is a single point by property (iii) in the definition of the class ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 13 A ( G ). Therefore F ∈ G f and since F ⊂ K it holds that cap ν ( F ) ≤ cap ν ( K ). This achievesthe proof. (cid:3) For any Γ ∈ G f and (cid:15) >
0, set (Γ) (cid:15) := { z ∈ D : dist( z, Γ) < (cid:15) } . We endow G f with theHausdorff metric, i.e., d H (Γ , Γ ) := inf { (cid:15) : Γ ⊂ (Γ ) (cid:15) , Γ ⊂ (Γ ) (cid:15) } . By standard properties of the Hausdorff distance [19, Sec. 3.16], clos d H ( G f ), the closureof G f in the d H -metric, is a compact metric space. Observe that taking d H -limit cannotincrease the number of connected components since any two components of the limit set havedisjoint (cid:15) -neighborhoods. That is, the d H -limit of a sequence of compact sets having lessthan N connected components has in turn less than N connected components. Moreover,each component of the d H -limit of a sequence of compact sets E n is the d H -limit of asequence of unions of components from E n . Thus, each element of clos d H ( G f ) still consistsof a finite number of continua each containing at least two points from E f but possibly withmultiply connected complement. However, the polynomial convex hull of such a set, that is,the union of the set with the bounded components of its complement, again belongs to G f unless the set touches T . Lemma 16.
Let
G ⊂ G f be such that each element of clos d H ( G ) is contained in D . Thenthe functional I ν [ · ] is finite and continuous on clos d H ( G ) .Proof. Let Γ ∈ clos d H ( G ) be fixed. Set (cid:15) := dist(Γ , T ) / > N (cid:15) (Γ ) := { Γ ∈ clos d H ( G ) : d H (Γ , Γ) < (cid:15) } . Then it holds that dist((Γ) (cid:15) , T ) ≥ (cid:15) for any Γ ∈ N (cid:15) (Γ ) and (cid:15) ≤ (cid:15) . Thus, the closure ofeach such (Γ) (cid:15) is at least (cid:15) away from T − (cid:15) .Let Γ ∈ N (cid:15) (Γ ) and set (cid:15) := d H (Γ , Γ). Denote by D and D the unbounded componentsof the complements of Γ and Γ, respectively. It follows from (7.24) that I ν [Γ ] is finite andthat I ν [Γ] − I ν [Γ ] = (cid:90) (cid:90) ( g D ( z, u ) − g D ( z, u )) d (cid:101) ν ∗ ( u ) d (cid:101) ν ∗ ( z ) , where (cid:101) ν ∗ is the balayage of ν ∗ onto T − (cid:15) . Since Γ ⊂ (Γ ) (cid:15) and Γ ⊂ (Γ) (cid:15) , g D ( · , u ) − g D ( · , u )is a harmonic function in G := C \ ((Γ) (cid:15) ∩ (Γ ) (cid:15) ) for each u ∈ G by the first claim in Section 7.3(recall that we agreed to continue g D ( · , u ) and g D ( · , u ) by zero outside of the closures of D and D , respectively). Thus, since Green functions are non-negative, we get from themaximum principle for harmonic functions and the fact that (cid:101) ν ∗ is a unit measure that | I ν [Γ] − I ν [Γ ] | ≤ max u ∈ T − (cid:15) max z ∈ ∂G | g D ( z, u ) − g D ( z, u ) | < max u ∈ T − (cid:15) (cid:18) max z ∈ ∂ (Γ) (cid:15) g D ( z, u ) + max z ∈ ∂ (Γ ) (cid:15) g D ( z, u ) (cid:19) . (6.2)Let γ be any connected component of Γ and G γ be the unbounded component of itscomplement. Observe that (Γ) (cid:15) = ∪ γ ( γ ) (cid:15) , where the union is taken over the (finitely many)components of Γ. Since D ⊂ G γ , we get that(6.3) g D ( z, u ) ≤ g G γ ( z, u )for any u ∈ D and z ∈ G γ \ u by the maximum principle. Set δ := (cid:112) (cid:15)/ cap( γ ) and L to be the log(1 + δ )-level line of g G γ ( · , ∞ ). As G γ is simplyconnected, L is a smooth Jordan curve. Since γ is a continuum, it is well-known thatcap( γ ) ≥ diam( γ ) / γ contains at least two points from E f . Thus, diam( γ ) is bounded from below by the minimal distance between the algebraicsingularities of f . Hence, we can assume without loss of generality that δ ≤
1. We claim thatdist( γ, L ) ≥ (cid:15) and postpone the proof of this claim until the end of this lemma. The claimimmediately implies that ( γ ) (cid:15) is contained in the bounded component of the complement of L and that(6.4) max z ∈ ∂ ( γ ) (cid:15) g G γ ( z, ∞ ) ≤ log(1 + δ ) ≤ δ. It follows from the conformal invariance of the Green function [45, Thm. 4.4.2] and canbe readily verified using the characteristic properties that g G γ ( z, u ) = g G uγ (1 / ( z − u ) , ∞ ),where G uγ is the image of G γ under the map 1 / ( · − u ). It is also simple to compute that(6.5) dist( γ u , ∂ ( γ ) u(cid:15) ) ≤ (cid:15) dist( u, γ )dist( u, ∂ ( γ ) (cid:15) ) ≤ (cid:15)(cid:15) , u ∈ T − (cid:15) , by the remark after (6.1), where γ u and ( γ ) u(cid:15) have obvious meaning. So, combining (6.5)with (6.4) applied to γ u , we deduce that(6.6) max z ∈ ∂ ( γ ) (cid:15) g G γ ( z, u ) = max z ∈ ∂ ( γ ) u(cid:15) g G uγ ( z, ∞ ) ≤ max z ∈ ∂ ( γ u ) (cid:15)/(cid:15) g G uγ ( z, ∞ ) ≤ δ u , u ∈ T − (cid:15) , where we put δ u := (cid:112) (cid:15)/(cid:15) cap( γ u ).As we already mentioned, cap( γ ) ≥ diam( γ ) /
4. Hence, it holds that(6.7) min u ∈ T − (cid:15) cap( γ u ) ≥
14 min u ∈ T − (cid:15) max z,w ∈ γ (cid:12)(cid:12)(cid:12)(cid:12) z − u − w − u (cid:12)(cid:12)(cid:12)(cid:12) ≥ diam( γ )16 . Gathering together (6.3), (6.6), and (6.7), we derive thatmax u ∈ T − (cid:15) max z ∈ ∂ (Γ) (cid:15) g D ( z, u ) ≤ max γ (cid:15) (cid:115) (cid:15) diam( γ ) , where γ ranges over all components of Γ. Recall that each component of Γ contains at leasttwo points from E f . Thus, 1 / diam( γ ) is bounded above by a constant that depends onlyon f .Arguing in a similar fashion for Γ , we obtain from (6.2) that | I ν [Γ] − I ν [Γ ] | ≤ const. (cid:15) (cid:112) d H (Γ , Γ ) for any Γ ∈ N (cid:15) (Γ ) , where const. is a constant depending only on f . This finishes the proof of the lemma grantedwe prove the claim made before (6.4).It was claimed that for a continuum γ and the log(1 + δ )-level line L of g G γ ( · , ∞ ), δ ≤ γ, L ) ≥ δ cap( γ )2 , where G γ is the unbounded component of the complement of γ . Inequality (6.8) was provedin [42, Lem. 1], however, this work was never published and the authors felt compelled toreproduce this lemma here. By conformal invariance of Green functions it is enough to check it for G γ = O in which case it isobvious. ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 15
Let Φ be a conformal map of O onto G γ , Φ( ∞ ) = ∞ . It is well-known that | Φ( z ) z − | → cap( γ ) as z → ∞ and that g G γ ( · , ∞ ) = log | Φ − | , where Φ − is the inverse of Φ (that is, aconformal map of G γ onto O , Φ − ( ∞ ) = ∞ ). Then it follows from [24, Thm. IV.2.1] that(6.9) | Φ (cid:48) ( z ) | ≥ cap( γ ) (cid:18) − | z | (cid:19) , z ∈ O . Let z ∈ γ and z ∈ L be such that dist( γ, L ) = | z − z | . Denote by [ z , z ] the segmentjoining z and z . Observe that Φ − maps the annular domain bounded by γ and L ontothe annulus { z : 1 < | z | < δ } . Denote by S the intersection of Φ − (( z , z )) with thisannulus. Clearly, the angular projection of S onto the real line is equal to (1 , δ ). Thendist( γ, L ) = (cid:90) ( z ,z ) | dz | = (cid:90) Φ − (( z ,z )) | Φ (cid:48) ( z ) || dz | ≥ cap( γ ) (cid:90) Φ − (( z ,z )) (cid:18) − | z | (cid:19) | dz |≥ cap( γ ) (cid:90) S (cid:18) − | z | (cid:19) | dz | ≥ cap( γ ) (cid:90) (1 , δ ) (cid:18) − | z | (cid:19) | dz | = δ cap( γ )1 + δ , where we used (6.9). This proves (6.8) since it is assumed that δ ≤ (cid:3) Set pr ρ ( · ) to be the radial projection onto D ρ , i.e., pr ρ ( z ) = z if | z | ≤ ρ and pr ρ ( z ) = ρz/ | z | if ρ < | z | < ∞ . Put further pr ρ ( K ) := { pr ρ ( z ) : z ∈ K } . In the following lemma we showthat pr ρ can only increase the value of I ν [ · ]. Lemma 17.
Let Γ ∈ G f and ρ ∈ [ r, , r = max z ∈ E f | z | . Then pr ρ (Γ) ∈ G f and cap ν (pr ρ (Γ)) ≤ cap ν (Γ) .Proof. As E f ⊂ D r , f naturally extends along any ray tξ , ξ ∈ T , t ∈ ( r, ∞ ). Thus, the germ f has a representative which is single-valued and meromorphic outside of pr ρ (Γ). It is alsotrue that pr ρ is a continuous map on C and therefore cannot disconnect the components ofΓ although it may merge some of them. Thus, pr ρ (Γ) ∈ G f .Set w = exp { U ν } and δ wm (Γ) := sup z ,...,z m ∈ Γ (cid:89) ≤ j ρ and | z | = | z | = x > ρ . In the former situation, (6.10) will follow upon showing that l ( x ) := x + | z | − x | z | cos φ x | u | − x | u | cos ψ is an increasing function on ( | z | , / | u | ) for any choice of φ and ψ . Since l (cid:48) ( x ) = 2 x (1 − | u | | z | ) − | z | cos φ (1 − x | u | ) − | u | cos ψ ( x − | z | )(1 + x | u | − x | u | cos ψ ) > − | u || z | )(1 − x | u | )( x − | z | )(1 + x | u | ) > , l is indeed strictly increasing on ( | z | , / | u | ). In the latter case, (6.10) is equivalent toshowing that l ( x ) := (1 /x + x | u | − | u | cos φ )(1 /x + x | u | − | u | cos ψ )is a decreasing function on ( ρ, / | u | ) for any choice of φ and ψ . This is true since l (cid:48) ( x ) = 2( | u | − /x )(1 /x + x | u | − | u | (cos φ + cos ψ )) < . Thus, we verified (6.10) for ν = δ u .In the general case it holds that | z − z | w ( z ) w ( z ) = exp (cid:26)(cid:90) log | z − z || − z ¯ u || − z ¯ u | dν ( u ) (cid:27) . As the kernel on the right-hand side of the equality above gets smaller when z j is replacedby pr ρ ( z j ), j = 1 ,
2, by what precedes, the validity of (6.10) follows. (cid:3)
Combining Lemmas 15–17, we obtain the existence of minimal sets.
Lemma 18.
A minimal set Γ ν exists and is contained in D r , r = max {| z | : z ∈ E f } .Proof. By Lemma 15, it is enough to consider only the sets in G f . Let { Γ n } ⊂ G f be amaximizing sequence for I ν [ · ] (minimizing sequence for the ν -capacity), that is, I ν [Γ n ] tendsto sup Γ ∈G f I ν [Γ] as n → ∞ . Then it follows from Lemma 17 that { pr r (Γ n ) } is anothermaximizing sequence for I ν [ · ] in G f , and pr r (Γ n ) ∈ G r := { Γ ∈ G f : Γ ⊆ D r } . Asclos d H ( G r ) is a compact metric space, there exists at least one limit point of { pr r (Γ n ) } inclos d H ( G r ), say Γ , and Γ ⊂ D r . Since I ν [ · ] is continuous on clos d H ( G r ) by Lemma 16, I ν [Γ ] = sup Γ ∈G f I ν [Γ]. Finally, as the polynomial convex hull of Γ , say Γ (cid:48) , belongs to G f and since I ν [Γ ] = I ν [Γ (cid:48) ] (see Section 7.2.4), we may put Γ ν = Γ (cid:48) . (cid:3) To continue with our analysis we need the following theorem [32, Thm. 3.1]. It describesthe continuum of minimal condenser capacity connecting finitely many given points as aunion of closures of the non-closed negative critical trajectories of a quadratic differential.Recall that a negative trajectory of the quadratic differential q ( z ) dz is a maximally con-tinued arc along which q ( z ) dz <
0; the trajectory is called critical if it ends at a zero or apole of q ( z )[32, 43]. Theorem K.
Let A = { a , . . . , a m } ⊂ D be a set of m ≥ distinct points. Then thereuniquely exists a continuum K , A ⊂ K ⊂ D , such that cap( K , T ) ≤ cap( K, T ) for any other continuum with A ⊂ K ⊂ D . Moreover, there exist m − points b , . . . , b m − ∈ D such that K is the union of the closures of the non-closed negative critical trajectories ofthe quadratic differential q ( z ) dz , q ( z ) := ( z − b ) · . . . · ( z − b m − )(1 − ¯ b z ) · . . . · (1 − ¯ b m − z )( z − a ) · . . . · ( z − a m )(1 − ¯ a z ) · . . . · (1 − ¯ a m z ) , contained in D . There exists only finitely many such trajectories. Furthermore, the equilib-rium potential V ω ( K , T ) D satisfies (cid:16) ∂ z V ω ( K , T ) D ( z ) (cid:17) = q ( z ) , z ∈ D . The last equation in Theorem K should be understood as follows. The left-hand side ofthis equality is defined in D \ K and represents a holomorphic function there, which coincideswith q on its domain of definition. As K has no interior because critical trajectories areanalytic arcs with limiting tangents at their endpoints [43], the equality on the whole set D ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 17 is obtained by continuity. Note also that D \ K is connected by unicity claimed in Theorem6.1, for the polynomial convex hull of K has the same Green capacity as K ( cf. section7.1.3). Moreover, it follows from the local theory of quadratic differentials that each b j isthe endpoint of at least three arcs of K (because b j is a zero of q ( z )) and that each a j isthe endpoint of exactly one arc of K (because a j is a simple pole of q ( z )).Having Theorem K at hand, we are ready to describe the structure of a minimal set Γ ν . Lemma 19.
A minimal set Γ ν is symmetric ( Definition 10 ) with respect to ν ∗ .Proof. Let (cid:101) ν ∗ be the balayage of ν ∗ onto T ρ with ρ < ν inthe interior of D ρ . Let γ be any of the continua constituting Γ ν . Clearly V := V (cid:101) ν ∗ D ν , where D ν = C \ Γ ν , is harmonic in D ν \ T ρ and extends continuously to the zero function on Γ ν since Γ ν is a regular set. Moreover, by Sard’s theorem on regular values [27, Sec. 1.7] thereexists δ > { z : V ( z ) < δ } containing γ , is itself contained in D ρ and its boundary is an analytic Jordan curve, say L . Let φ bea conformal map of Ω onto D . Set (cid:101) γ := φ − ( (cid:101) K ), where (cid:101) K is the continuum of minimalcondenser capacity for φ ( E f ∩ γ ). Our immediate goal is to show that γ = (cid:101) γ .Assume to the contrary that γ (cid:54) = (cid:101) γ , i.e., φ ( γ ) =: K (cid:54) = (cid:101) K , and therefore(6.11) cap( (cid:101) K, T ) < cap( K, T ) . Set(6.12) (cid:101) V := (cid:40) δ cap( (cid:101) K, T ) (cid:104) V ω ( T , (cid:102) K ) C \ (cid:101) K ◦ φ (cid:105) , z ∈ Ω ,V, z / ∈ Ω , where ω ( T , (cid:101) K ) is the Green equilibrium distribution on T relative to C \ (cid:101) K . The functions V and (cid:101) V are continuous in Ω and equal to δ on L . Furthermore, they are harmonic in Ω \ γ and Ω \ (cid:101) γ and equal to zero on γ and (cid:101) γ , respectively. Then it follows from Lemma 24 andthe conformal invariance of the condenser capacity (7.7) that(6.13) 12 π (cid:90) L ∂V∂ n ds = − δ cap( K, T ) and 12 π (cid:90) L ∂ (cid:101) V∂ n ds = − δ cap( (cid:101) K, T ) , where ∂/∂ n stands for the partial derivative with respect to the inner normal on L . (InLemma 24, L should be contained within the domain of harmonicity of V and (cid:101) V . As V and (cid:101) V are constant on L , they can be harmonically continued across by reflection. Thus,Lemma 24 does apply.) Moreover, (cid:101) V − V (cid:101) ν ∗ (cid:101) D is a continuous function on C that is harmonicin (cid:101) D \ L by the first claim in Section 7.3, where (cid:101) D := ( D ν ∪ γ ) \ (cid:101) γ , and is identically zero onΓ := C \ (cid:101) D . Thus, we can apply Lemma 23 with (cid:101) V − V (cid:101) ν ∗ (cid:101) D and (cid:101) D (smoothness properties of V − V (cid:101) ν ∗ (cid:101) D follow from the fact that (cid:101) V can be harmonically continued across L ), which statesthat(6.14) (cid:101) V = V (cid:101) ν ∗ − σ (cid:101) D , dσ := 12 π ∂ ( (cid:101) V − V ) ∂ n ds, where σ is a finite signed measure supported on L (observe that the outer and inner normalderivatives of V (cid:101) ν ∗ (cid:101) D on L are opposite to each other as V (cid:101) ν ∗ (cid:101) D is harmonic across L and thereforethey do not contribute to the density of σ ; due to the same reasoning the outer normal In other words, if we put φ ( E f ∩ γ ) = { p , . . . , p m } and g ( z ) := 1 / m (cid:112)(cid:81) ( z − p j ), then (cid:101) K is the set ofminimal condenser capacity for g as defined in Definition 2. derivative of (cid:101) V is equal to minus the inner normal derivative of V by (6.12)). Hence, onecan easily deduce from (6.13) and (6.11) that(6.15) σ ( L ) = δ (cid:16) cap( K, T ) − cap( (cid:101) K, T ) (cid:17) > . Since the components of Γ ν and Γ contain exactly the same branch points of f and Γ hasconnected complement (for D ν is connected and so is C \ (cid:101) γ because D \ (cid:101) K is connected), itfollows that Γ ∈ G f by the monodromy theorem. Moreover, we obtain from (7.24), (6.12),and (6.14) that I ν [Γ] − I ν [Γ ν ] = I (cid:101) D [ (cid:101) ν ∗ ] − I D ν [ (cid:101) ν ∗ ] = (cid:90) (cid:16) V (cid:101) ν ∗ (cid:101) D − V (cid:17) d (cid:101) ν ∗ = (cid:90) V σ (cid:101) D d (cid:101) ν ∗ since supp( (cid:101) ν ∗ ) ∩ Ω = ∅ . Further, applying the Fubini-Tonelli theorem and using (6.14) oncemore, we get that I ν [Γ] − I ν [Γ ν ] = (cid:90) V (cid:101) ν ∗ (cid:101) D dσ = (cid:90) (cid:101) V dσ + I (cid:101) D [ σ ] = δσ ( L ) + I (cid:101) D [ σ ] > I ν [Γ ν ] is maximal among all sets in G f and therefore γ = (cid:101) γ . Hence, K = (cid:101) K = φ ( γ )and (cid:101) V = V .Observe now that by Theorem K stated just before this lemma and the remarks thereafter,the set K consists of a finite number of open analytic arcs and their endpoints. These fallinto two classes a , . . . , a m and b , . . . , b m − , members of the first class being endpoints ofexactly one arc and members of the second class being endpoints of at least three arcs. Thus,the same is true for γ . Moreover, the jump of f across any open arc C ⊂ γ cannot vanish,otherwise excising out this arc would leave us with an admissible compact set Γ (cid:48) ⊂ Γ ν ofstrictly smaller ν -capacity since ω Γ ν , − U ν ( C ) > ν is a smooth cut (Definition 3). Finally, we havethat ∂V∂ n ± γ = δ cap( φ ( γ ) , T ) (cid:18) ∂∂ n ± K V ω ( T ,K ) C \ K (cid:19) | φ (cid:48) | by (6.12) and the conformality of φ , where ∂/∂ n ± γ and ∂/∂ n ± K are the partial derivativeswith respect to the one-sided normals at the smooth points of γ and K , respectively. Thus,it holds that ∂V∂ n − γ = ∂V∂ n + γ on the open arcs constituting γ since the corresponding property holds for V ω ( T ,K ) C \ K by (2.5).As γ was arbitrary continuum from Γ ν , we see that all the requirements of Definition 10 arefulfilled. (cid:3) To finish the proof of Theorem 12, it only remains to show uniqueness of Γ ν , which isachieved through the following lemma: Lemma 20. Γ ν is uniquely characterized as a compact set symmetric with respect to ν ∗ .Proof. Let Γ s ∈ G f be symmetric with respect to ν ∗ and Γ ν be any set of minimal capacityfor Problem ( f, ν ). Such a set exists by Lemma 18 and it is symmetric by Lemma 19.Suppose to the contrary that Γ s (cid:54) = Γ ν , that is,(6.16) Γ s ∩ ( C \ Γ ν ) (cid:54) = ∅ ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 19 (Γ s cannot be a strict subset of Γ ν for it would have strictly smaller ν -capacity as pointedout in the proof of Lemma 19). We want to show that (6.16) leads to(6.17) I ν [Γ s ] − I ν [Γ ν ] > . Clearly, (6.17) is impossible by the very definition of Γ ν and therefore the lemma will beproven.By the very definition of symmetry (Definition 10), Γ ν and Γ s are smooth cuts for f . Inparticular, C \ Γ ν , C \ Γ s are connected and we have a decomposition of the formΓ s = E s ∪ E s ∪ (cid:91) γ sj and Γ ν = E ν ∪ E ν ∪ (cid:91) γ νj , where E s , E ν ⊆ E f , γ νj , γ sj are open analytic arcs, and each element of E s , E ν is an endpointof exactly one arc from (cid:83) γ sj , (cid:83) γ νj while E s , E ν are finite sets of points each elements ofwhich serving as an endpoint for at least three arcs from (cid:83) γ sj , (cid:83) γ νj , respectively. Moreover,the continuations of f from infinity that are meromorphic outside of Γ s and Γ ν , say f s and f ν , are such that the jumps f + s − f − s and f + ν − f − ν do not vanish on any subset with a limitpoint of (cid:83) γ sj and (cid:83) γ νj , respectively. Note that Γ s ∩ Γ ν (cid:54) = ∅ otherwise C \ (Γ ν ∪ Γ s ) wouldbe connected, so f could be continued analytically over ( C \ Γ ν ) ∪ ( C \ Γ s ) = C and it wouldbe identically zero by our normalization.Write Γ s = Γ s ∪ Γ s and Γ ν = Γ ν ∪ Γ ν , where Γ ks (resp. Γ kν ) are compact disjoint sets suchthat each connected component of Γ s (resp. Γ ν ) has nonempty intersection with Γ ν (resp.Γ s ) while Γ s ∩ Γ ν = Γ ν ∩ Γ s = ∅ .Now, put, for brevity, D ν := C \ Γ ν and D s := C \ Γ s . Denote further by Ω the unboundedcomponent of D ν ∩ D s . Then(6.18) Ω ∩ E s ∩ Γ s = Ω ∩ E ν ∩ Γ ν . Indeed, assume that there exists e ∈ (Ω ∩ E s ∩ Γ s ) \ ( E ν ∩ Γ ν ) and let γ se be the arc in the union (cid:83) γ sj that has e as one of the endpoints. By our assumption there is an open disk W centeredat e such that W ∩ Γ s = { e } ∪ ( W ∩ γ se ) and W ∩ Γ ν = ∅ . Thus W \ ( { e } ∪ γ se ) ⊂ D ν ∩ D s .Anticipating the proof of Proposition 11 in Section 6.2 (which is independent of the presentproof), γ se has well-defined tangent at e so we can shrink W to ensure that ∂W ∩ γ se isa single point. Then W \ ( { e } ∪ γ se ) is connected hence contained in a single connectedcomponent of D ν ∩ D s which is necessarily Ω since e ∈ Ω. As f s and f ν coincide on Ω and f ν is meromorphic in W , f s has identically zero jump on γ se ∩ W which is impossible by thedefinition of a smooth cut. Consequently the left hand side of (6.18) is included in the righthand side and the opposite inclusion can be shown similarly. ΩΓ s Γ s Γ s Γ ν Figure 1.
A particular example of Γ s (solid lines) and Γ ν (dashed lines). Blackdots represent branch points (black dots within big gray disk are branch point f that lie on other sheets of the Riemann surface than the one we fixed). The whitearea on the figure represents domain Ω. Next, observe that(6.19) Γ s ∩ Ω = ∅ . Indeed, since ∂ Ω ⊂ Γ s ∪ Γ ν and Γ s , Γ s ∪ Γ ν are disjoint compact sets, a connected componentof ∂ Ω that meets Γ s is contained in it. If z ∈ Γ s ∩ ∂ Ω lies on γ sj , then by analyticity of thelatter each sufficiently small disk D z centered at z is cut out by γ sj ∩ D z into two connectedcomponents included in D ν ∩ D s , and of necessity one of them is contained in Ω. Hence γ sj ∩ D z is contained in ∂ Ω, and in turn so does the entire arc γ sj by connectedness. Henceevery component of Γ s ∩ ∂ Ω consists of a union of arcs γ sj connecting at their endpoints.Because Γ s has no loop, one of them has an endpoint z ∈ E s ∪ E s belonging to no otherarc. If z ∈ E s , reasoning as we did to prove (6.18) leads to the absurd conclusion that f s has zero jump across the initial arc. If z ∈ E s , anticipating the proof of Proposition 11 onceagain, each sufficiently small disk D z centered at z is cut out by Γ s ∩ D z into curvilinearsectors included in D ν ∩ D s , and of necessity one of them is contained in Ω whence at leasttwo adjacent arcs γ sj emanating from z are included in ∂ Ω. This contradicts the fact that z belongs to exactly one arc of the hypothesized component of Γ s ∩ ∂ Ω, and proves (6.19).Finally, setΓ s := (cid:2) Γ s \ ( ∂ Ω \ E s ) (cid:3) ∩ D ν and Γ s := (cid:104) Γ s ∩ (cid:91) γ sj (cid:105) ∩ ∂ Ω ∩ D ν . Clearly(6.20) (cid:0) Γ s \ E s (cid:1) ∩ Ω = ∅ . Moreover, observing that any two arcs γ sj , γ νk either coincide or meet in a (possibly empty)discrete set and arguing as we did to prove (6.19), we see that (cid:2) Γ s ∩ (cid:83) γ sj (cid:3) ∩ ∂ Ω consistsof subarcs of arcs γ sj whose endpoints either belong to some intersection γ sj ∩ γ νk (in whichcase they contain this endpoint) or else lie in E s ∪ E s (in which case they do not containthis endpoint). Thus Γ s is comprised of open analytic arcs (cid:101) γ s(cid:96) contained in ∂ Ω ∩ (cid:83) γ sj anddisjoint from Γ ν . Hence for any z ∈ Γ s , say z ∈ (cid:101) γ s(cid:96) , and any disk D z centered at z of smallenough radius it holds that D z ∩ ∂ Ω = D z ∩ (cid:101) γ s(cid:96) and that D z \ (cid:101) γ s(cid:96) has exactly two connectedcomponents:(6.21) D z ∩ Ω (cid:54) = ∅ and D z ∩ (cid:0) C \ Ω (cid:1) (cid:54) = ∅ for if z ∈ (cid:101) γ s(cid:96) was such that D z \ (cid:101) γ s(cid:96) ⊂ Ω, the jump of f s across (cid:101) γ s(cid:96) would be zero as the jumpof f ν is zero there and f s = f ν in Ω (see Figure 1).As usual, denote by (cid:101) ν ∗ the balayage of ν ∗ onto T ρ with ρ ∈ ( r,
1) but large enough sothat Γ s and Γ ν are contained in the interior of D ρ (see Lemma 18 for the definition of r ).Then, according to (7.24) and (7.39), it holds that(6.22) I ν [Γ s ] − I ν [Γ ν ] = I D s [ (cid:101) ν ∗ ] − I D ν [ (cid:101) ν ∗ ] = D D s ( V s ) − D D ν ( V ν ) , where V s := V (cid:101) ν ∗ D s and V ν := V (cid:101) ν ∗ D ν . Indeed, as (cid:101) ν ∗ has finite energy (see Section 7.2.3), theDirichlet integrals of V s and V ν in the considered domains (see Section 7.4) are well-definedby Proposition 11, which is proven later but independently of the results in this section.Set D := D ν \ (Γ s ∪ Γ s ). Since (cid:2) Γ s \ ( ∂ Ω \ E s ) (cid:3) consists of piecewise smooth arcs in Γ s whose endpoints either belong to this arc (if they lie in E s ), or to E s ∩ Γ s (hence also toΓ ν by (6.18)), or else to some intersection γ sj ∩ γ νk (in which case they belong to Γ ν again),we see that D is an open set. As V ν is harmonic across Γ s ∪ Γ s and V s is harmonic acrossΓ ν \ Γ s , we get from (7.38) that(6.23) D D ν ( V ν ) = D D ( V ν ) and D D s ( V s ) = D D \ Γ s ( V s ) ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 21 since D s \ Γ ν = D ν \ Γ s = D \ Γ s , by inspection on using (6.18).Now, recall that Γ s has no interior and V s ≡ s , that is, V s is defined in the wholecomplex plane. So, we can define a function on C by putting(6.24) (cid:101) V := (cid:26) V s , in Ω , − V s , otherwise . We claim that (cid:101) V is superharmonic in D and harmonic in D \ T ρ . Indeed, it is clearlyharmonic in D \ (Γ s ∪ T ρ ) = ( D s ∩ D ν ) \ T ρ and superharmonic in a neighborhood of T ρ ⊂ Ωwhere its weak Laplacian is − π (cid:101) ν ∗ which is a negative measure. Moreover, Γ s is a collectionof open analytic arcs such that ∂V s /∂ n + = ∂V s /∂ n − by the symmetry of Γ s , where n ± are the two-sided normal on each subarc of Γ s . The equality of the normals means that V s can be continued harmonically across each subarc of Γ s by − V s . Hence, (6.21) and thedefinition of (cid:101) V yield that it is harmonic across Γ s thereby proving the claim. Thus, using(7.41) (applied with D (cid:48) = Ω) and (7.38), we obtain(6.25) D D \ Γ s ( V s ) = D D \ Γ s ( (cid:101) V ) = D D ( (cid:101) V )hence combining (6.22), (6.23), and (6.25), we see that(6.26) I ν [Γ s ] − I ν [Γ ν ] = D D ( (cid:101) V ) − D D ( V ν ) . By the first claim in Section 7.3, it holds that h := (cid:101) V − V ν is harmonic in D . Observethat h is not a constant function, for it tends to zero at each point of Γ s ∩ Γ ν ⊂ ∂D whereasit tends to a strictly negative value at each point of Γ s ∩ D ν ⊂ D which is nonempty by(6.16). Then(6.27) D D ( (cid:101) V ) = D D ( V ν ) + D D ( h ) + 2 D D ( V ν , h ) . Now, V ν ≡ ν and it is harmonic across Γ s ∪ Γ s , hence ∂h∂ n + + ∂h∂ n − = ∂ (cid:101) V∂ n + + ∂ (cid:101) V∂ n − on Γ s ∪ Γ s . Consequently, we get from (7.35), since (cid:101) V = − V s in the neighborhood of Γ s ∪ Γ s by (6.19)and (6.20), that(6.28) D D ( V ν , h ) = − (cid:90) Γ s ∪ Γ s V ν (cid:32) ∂ (cid:101) V∂ n + + ∂ (cid:101) V∂ n − (cid:33) ds π = (cid:90) Γ s ∪ Γ s V ν (cid:18) ∂V s ∂ n + + ∂V s ∂ n − (cid:19) ds π ≥ V ν is nonnegative while ∂V s /∂n + , ∂V s /∂n − are also nonnegative on Γ s ∪ Γ s as V s ≥ I ν [Γ s ] − I ν [Γ ν ] ≥ D D ( h ) > h = ˜ V − V ν is a non-constant harmonic function in D . This shows (6.17)and finishes the proof of the lemma. (cid:3) Proof of Proposition 11.
It is well known that H ω, Γ is holomorphic in the domainof harmonicity of V ω C \ Γ , that is, in C \ (Γ ∪ supp( ω )). It is also clear that H ± ω, Γ exist smoothlyon each γ j since V ω C \ Γ can be harmonically continued across each side of γ j .Denote by n ± t the one-sided unit normals at t ∈ (cid:83) γ j and by τ t the unit tangent pointing inthe positive direction. Let further n ± ( t ) be the unimodular complex numbers corresponding to vectors n ± t . Then the complex number corresponding to τ t is ∓ in ± ( t ) and it can bereadily verified that ∂V ω C \ Γ ∂ n ± t = 2Re (cid:16) n ± ( t ) H ± ω, Γ ( t ) (cid:17) and ∂ (cid:16) V ω C \ Γ (cid:17) ± ∂τ t = ∓ (cid:16) n ± ( t ) H ± ω, Γ ( t ) (cid:17) . As (cid:16) V ω C \ Γ (cid:17) ± ≡ n ± H ± ω, Γ is real on Γ. Moreover since n + = − n − and by the symmetry property (4.3), it holds that H + ω, Γ = − H − ω, Γ on (cid:83) γ j . Hence, H ω, Γ is holomorphic in C \ ( E ∪ E ∪ supp( ω )). Since E ∪ E consists of isolated points around which H ω, Γ is holomorphic each e ∈ E ∪ E is either apole, a removable singularity, or an essential one. As H ω, Γ is holomorphic on a two-sheetedRiemann surface above the point, it cannot have an essential singularity since its primitivehas bounded real part ± V ω C \ Γ . Now, by repeating the arguments in [43, Sec. 8.2], we deducethat ( z − e ) j e − H ω, Γ ( z ) is holomorphic and non-vanishing in some neighborhood of e where j e is the number of arcs γ j having e as an endpoint, that the tangents at e to these arcsexist, and that they are equiangular if j e > Proof of Theorem 14.
The following theorem [25, Thm. 3] and its proof are essen-tial in establishing Theorem 14. Before stating this result, we remind the reader that apolynomial v is said to by spherically normalized if it has the form(6.29) v ( z ) = (cid:89) v ( e )=0 , | e |≤ ( z − e ) (cid:89) v ( e )=0 , | e | > (1 − z/e ) . We also recall from [25] the notions of a tame set and a tame point of a set. A point z belonging to a compact set Γ is called tame, if there is a disk centered at z whose intersectionwith Γ is an analytic arc. A compact set Γ is called tame, if Γ is non-polar and quasi-everypoint of Γ is tame.A tame compact set Γ is said to have the S-property in the field ψ , assumed to beharmonic in some neighborhood of Γ, if supp( ω Γ ,ψ ) forms a tame set as well, every tamepoint of supp( ω Γ ,ψ ) is also a tame point of Γ, and the equality in (4.4) holds at each tamepoint of supp( ω Γ ,ψ ).Whenever the tame compact set Γ has connected complement in a simply connectedregion G ⊃ Γ and g is holomorphic in G \ Γ, we write (cid:72) Γ g ( t ) dt for the contour integral of g over some (hence any) system of curves encompassing Γ once in G in the positive direction.Likewise, the Cauchy integral (cid:72) Γ g ( t ) / ( z − t ) dt can be defined at any z ∈ C \ Γ by choosingthe previous system of curves in such a way that it separates z from Γ.If g has limits from each side at tame points of Γ, and if these limits are integrablewith respect to linear measure on Γ, then the previous integrals may well be rewritten asintegrals on Γ with g replaced by its jump across Γ. However, this is not what is meant bythe notation (cid:72) Γ . Theorem GR.
Let G ⊂ D be a simply connected domain and Γ ⊂ G be a tame compactset with connected complement. Let also g be holomorphic in G \ Γ and have continuouslimits on Γ from each side in the neighborhood of every tame point, whose jump across Γ isnon-vanishing q.e. Further, let { Ψ n } be a sequence of functions that satisfy: (1) Ψ n is holomorphic in G and − n log | Ψ n | → ψ locally uniformly there, where ψ isharmonic in G ; (2) Γ possesses the S-property in the field ψ ( see (4.4)) . ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 23
Then, if the polynomials q n , deg( q n ) ≤ n , satisfy the orthogonality relations (6.30) (cid:73) Γ q n ( t ) l n − ( t )Ψ n ( t ) g ( t ) dt = 0 , for any l n − ∈ P n − , then µ n ∗ → ω Γ ,ψ , where µ n is the normalized counting measure of zeros of q n . Moreover, ifthe polynomials q n are spherically normalized, it holds that (6.31) | A n ( z ) | / n cap → exp {− c ( ψ ; Γ) } in C \ Γ , where c ( ψ ; Γ) is the modified Robin constant ( Section 7.1.2 ) , and (6.32) A n ( z ) := (cid:73) Γ q n ( t ) (Ψ n g )( t ) dtz − t = q n ( z ) l n ( z ) (cid:73) Γ ( l n q n )( t ) (Ψ n g )( t ) dtz − t , where l n can be any nonzero polynomial of degree at most n .Proof of Theorem 14. Let E n be the sets constituting the interpolation scheme E . Set Ψ n tobe the reciprocal of the spherically normalized polynomial with zeros at the finite elementsof E n , i.e., Ψ n = 1 / ˜ v n , where ˜ v n is the spherical renormalization of v n (see Definition 13and (6.29)). Then the functions Ψ n are holomorphic and non-vanishing in C \ supp( E ) (inparticular, in D ), n log | Ψ n | cap → U ν in C \ supp( ν ∗ ) by Lemma 21, and this convergence islocally uniform in D by definition of the asymptotic distribution and since log 1 / | z − t | iscontinuous on a neighborhood of supp( E ) for fixed z ∈ D . As U ν is harmonic in D , require-ment (1) of Theorem GR is fulfilled with G = D and ψ = − U ν . Further, it follows fromTheorem 12 that Γ ν is a symmetric set. In particular it is a smooth cut, hence it is tamewith tame points ∪ j γ j . Moreover, since Γ ν is regular, we have that supp( ω Γ ,ψ ) = Γ ν by(7.18) and properties of balayage (Section 7.2.2). Thus, by the remark after Definition 10,symmetry implies that Γ ν possesses the S-property in the field − U ν and therefore require-ment (2) of Theorem GR is also fulfilled. Let now Q , deg( Q ) =: m , be a fixed polynomialsuch that the only singularities of Qf in D belong to E f . Then Qf is holomorphic andsingle-valued in D \ Γ ν , it extends continuously from each side on ∪ γ j , and has a jump therewhich is continuous and non-vanishing except possibly at countably many points. All therequirement of Theorem GR are then fulfilled with g = Qf .Let L ⊂ D be a smooth Jordan curve that separates Γ ν and the poles of f (if any) from E .Denote by q n the spherically normalized denominators of the multipoint Pad´e approximantsto f associated with E . It is a standard consequence of Definition 13 (see e.g. [25, sec.1.5.1]) that(6.33) (cid:90) L z j q n ( z )Ψ n ( z ) f ( z ) dz = 0 , j ∈ { , . . . , n − } . Clearly, relations (6.33) imply that(6.34) (cid:73) Γ ν ( lq n Ψ n f Q )( t ) dt = 0 , deg( l ) < n − m. Equations (6.34) differ from (6.30) only in the reduction of the degree of polynomials l bya constant m . However, to derive the first conclusion of Theorem GR, namely that µ n ∗ → ω Γ ,ψ , orthogonality relations (6.30) are used solely when applied to a specially constructedsequence { l n } such that l n = l n, l n, , where deg( l n, ) ≤ nθ , θ <
1, and deg( l n, ) = o ( n ) as n → ∞ (see the proof of [25, Thm. 3] in between equations (27) and (28)). Thus, the proof Note that the orthogonality in (6.30) is non-Hermitian, that is, no conjugation is involved. The fact that we can pick an arbitrary polynomial l n for this integral representation of A n is a simpleconsequence of orthogonality relations (6.30). is still applicable in our situation, to the effect that the normalized counting measures ofthe zeros of q n converge weak ∗ to (cid:98) ν ∗ = ω Γ ν , − U ν , see (7.23).For each n ∈ N , let q n,m , deg( q n,m ) = n − m , be a divisor of q n . Observe that thepolynomials q n,m have exactly the same asymptotic zero distribution in the weak ∗ sense asthe polynomials q n . Put(6.35) A n,m ( z ) := (cid:73) ( q n,m q n )( t ) (Ψ n f Q )( t ) dtz − t , z ∈ D ν . Due to orthogonality relations (6.34), A n,m can be equivalently rewritten as(6.36) A n,m ( z ) := q n,m ( z ) l n − m ( z ) (cid:73) ( l n − m q n )( t ) (Ψ n f Q )( t ) dtz − t , z ∈ D ν , where l n − m is an arbitrary polynomial of degree at most n − m . Formulae (6.35) and (6.36)differ from (6.32) in the same manner as orthogonality relations (6.34) differ from those in(6.30). Examination of the proof of [25, Thm. 3] (see the discussion there between equations(33) and (37)) shows that limit (6.31) is proved using expression (6.32) for A n with a choiceof polynomials l n that satisfy some set of asymptotic requirements and can be chosen tohave the degree n − m . Hence it still holds that(6.37) | A n,m ( z ) | / n cap → exp {− c ( − U ν ; Γ ν ) } in D ν . Finally, using the Hermite interpolation formula like in [52, Lem. 6.1.2], the error ofapproximation has the following representation(6.38) ( f − Π n )( z ) = A n,m ( z )( q n,m q n Q Ψ n )( z ) , z ∈ D ν . From Lemma 21 we know that log(1 / | q n | ) /n cap → V (cid:99) ν ∗ ∗ = V (cid:99) ν ∗ in D ν , since ordinary andspherically normalized potentials coincide for measures supported in D . This fact togetherwith (6.37) and (6.38) easily yield that | f − Π n | / n cap → exp (cid:110) − c ( − U ν ; Γ ν ) + V (cid:98) ν ∗ − U ν (cid:111) in D ν \ supp( ν ∗ ) . Therefore, (5.1) follows from (7.20) and the fact that U ν = V ν ∗ ∗ by the remark at thebeginning of Section 7.2.4. (cid:3) Proof of Theorem 4, Theorem 7, Corollary 8, Theorem 6, and Theorem 5.
Proof of Theorem 4.
Let Γ ∈ K f ( G ) be a smooth cut for f that satisfies (2.5) and Θ be aconformal map of D onto G . Set K := Θ − (Γ). Then we get from the conformal invarianceof the condenser capacity (see (7.7)) and the maximum principle for harmonic functions thatcap(Γ , T ) = cap( K, T ) and V ω ( K, T ) C \ K = V ω (Γ ,T ) C \ Γ ◦ Θ in D . As Θ is conformal in D , it can be readily verified that V ω ( K, T ) C \ K satisfies (2.5) as well (naturally,on K ). Univalence of Θ also implies that the continuation properties of ( f ◦ Θ)(Θ (cid:48) ) / in D are exactly the same as those of f in G . Moreover, this is also true for f Θ , the orthogonalprojection of ( f ◦ Θ)(Θ (cid:48) ) / from L onto ¯ H (see Section 3). Indeed, f Θ is holomorphicin O by its very definition and can be continued analytically across T by ( f ◦ Θ)(Θ (cid:48) ) / minus the orthogonal projection of the latter from L onto H , which is holomorphic in D by definition. Thus, f Θ ∈ A ( D ) and Γ ∈ K f ( G ) if and only if K ∈ K f Θ . Therefore, it isenough to consider only the case G = D . ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 25
Let Γ ∈ K f be a smooth cut for f that satisfies (2.5) and K be the set of minimal condensercapacity ( cf. Theorem 2). We must prove that Γ = K. Set, for brevity, D Γ := C \ Γ, V Γ := V ω ( T , Γ) D Γ , D K := C \ K, V K := V ω ( T , K) D K , and Ω to be the unbounded component of D K ∩ D Γ . Let also f D Γ and f D K indicate the meromorphic branches of f in D Γ and D K ,respectively. Arguing as we did to prove (6.19), we see that no connected component of ∂ Ωcan lie entirely in Γ \ K (resp. K \ Γ) otherwise the jump of f D Γ (resp. f D K ) across somesubarc of Γ (resp. K) would vanish. Hence by connectedness(6.39) Γ ∩ K ∩ ∂ Ω (cid:54) = ∅ . First, we deal with the special situation where ω ( T , K) = ω ( T , Γ) . Then V Γ − V K is harmonicin Ω by the first claim in Section 7.3. As both potentials are constant in O ⊂ Ω, we getthat V Γ = V K + const. in Ω. Since K and Γ are regular sets, potentials V Γ and V K extendcontinuously to ∂ Ω and vanish at ∂ Ω ∩ Γ ∩ K which is non-empty by (6.39). Thus, equalityof the equilibrium measures means that V Γ ≡ V K in Ω. However, because V Γ (resp. V K )vanishes precisely on Γ (resp. K), this is possible only if ∂ Ω ⊂ Γ ∩ K. Taking complementsin C , we conclude that D Γ ∪ D K , which is connected and contains ∞ , does not meet ∂ Ω.Therefore D Γ ∪ D K ⊂ Ω ⊂ D Γ ∩ D K , hence D Γ = D K thus Γ = K, as desired.In the rest of the proof we assume for a contradiction that Γ (cid:54) = K. Then ω ( T , K) (cid:54) = ω ( T , Γ) in view of what precedes, and therefore(6.40) D D K ( V K ) = I D K (cid:2) ω ( T , K) (cid:3) < I D K (cid:2) ω ( T , Γ) (cid:3) = D D K ( V ω ( T , Γ) D K )by (7.39) and since the Green equilibrium measure is the unique minimizer of the Greenenergy.The argument now follows the lines of the proof of Lemma 20. Namely, we writeΓ = E Γ0 ∪ E Γ1 ∪ (cid:91) γ Γ j , K = E K0 ∪ E K1 ∪ (cid:91) γ K j , and we define the sets Γ , Γ , Γ , Γ like we did in that proof for Γ s , Γ s , Γ s , Γ s , uponreplacing D s by D Γ , D ν by D K , E sj by E Γ j , E νj by E K j , γ sj by γ Γ j and γ νj by γ K j . The samereasoning that led to us to (6.19) and (6.20) yields(6.41) Γ ∩ Ω = ∅ , (cid:0) Γ \ E Γ1 (cid:1) ∩ Ω = ∅ . Subsequently we set D := D K \ (Γ ∪ Γ ) and we prove in the same way that it is an openset satisfying(6.42) D D K ( V ω ( T , Γ) D K ) = D D ( V ω ( T , Γ) D K ) and D D Γ ( V Γ ) = D D \ Γ ( V Γ )(compare (6.23)). Defining (cid:101) V as in (6.24) with V s replaced by V Γ , and using the symmetryof Γ (that is, (2.5) with Γ instead of K, which allows us to continue V Γ harmonically by − V Γ across each arc γ Γ j ) we find that (cid:101) V is harmonic in D \ T , superharmonic in D , and that(6.43) D D Γ \ Γ ( V Γ ) = D D ( (cid:101) V )(compare (6.25)). Next, we set h := (cid:101) V − V ω ( T , Γ) D K which is harmonic in D by the first claimin Section 7.3, and since h = V Γ − V ω (Γ , T ) D K in Ω ⊃ T . Because (cid:101) V = − V Γ in the neighborhoodof Γ s ∪ Γ s by (6.41), the same computation as in (6.28) gives us D D ( V ω ( T , Γ) D K , h ) ≥ , so we get from (7.39), (6.42), (6.43), (7.40) and (6.40) that I D Γ [ ω ( T , Γ) ] = D D Γ ( V Γ ) = D D ( (cid:101) V ) = D D ( V ω ( T , Γ) D K + h )= D D ( V ω ( T , Γ) D K ) + 2 D D ( V ω ( T , Γ) D K , h ) + D D ( h ) ≥ D D K ( V ω ( T , Γ) D K ) + D D ( h ) > D D K ( V K ) = I D K [ ω ( T , K) ] . (6.44)However, it holds that I D K [ ω ( T , K) ] = 1 / cap(K , T ) and I D Γ [ ω ( T , Γ) ] = 1 / cap(Γ , T )by (7.6). Thus, (6.44) yields that cap(Γ , T ) < cap(K , T ), which is impossible by the verydefinition of K. This contradiction finishes the proof. (cid:3) Proof of Theorem 7.
Let { r n } be a sequence of irreducible critical points for f . Further, let ν n be the normalized counting measures of the poles of r n and ν be a weak ∗ limit pointof { ν n } , i.e., ν n ∗ → ν , n ∈ N ⊂ N . Recall that all the poles of r n are contained in D andtherefore supp( ν ) ⊆ D .By Theorem 12, there uniquely exists a minimal set Γ ν for Problem ( f, ν ). Let Z n bethe set of poles of r n , where each pole appears with twice its multiplicity. As mentioned inSection 3, each r n interpolates f at the points of Z ∗ n , counting multiplicity. Hence, { r n } n ∈ N is the sequence of multipoint Pad´e approximants associated with the triangular scheme E = { Z ∗ n } n ∈ N that has asymptotic distribution ν ∗ , where ν ∗ is the reflection of ν across T .So, according to Theorem 14 (applied for subsequences), it holds that ν = (cid:98) ν ∗ , supp( ν ) = Γ ν ,i.e., ν is the balayage of its own reflection across T relative to D ν .Applying Lemma 25, we deduce that ν is the Green equilibrium distribution on Γ ν relativeto D , that is, ν = ω (Γ ν , T ) , and (cid:101) ν , the balayage of ν onto T , is the Green equilibriumdistribution on T relative to D ν , that is, (cid:101) ν = ω ( T , Γ ν ) . Moreover, Lemma 25 yields that V ν ∗ D ν = V (cid:101) νD ν in D and therefore V (cid:101) νD ν enjoys symmetry property (2.5) by Theorem 12. Hence,we get from Theorem 4 that Γ ν = K, the set of minimal condenser capacity for f , and that ν = ω (K , T ) . Since ν was an arbitrary limit point of { ν n } , we have that ν n ∗ → ω (K , T ) as n → ∞ . Finally, observe that (3.2) is a direct consequence of Theorem 14.To prove (3.3), we need to go back to representation (6.38), where q n ∈ M n is thedenominator of an irreducible critical point r n and q n,m , deg( q n,m ) = n − m , is an arbitrarydivisor of q n , while Ψ n = 1 / (cid:101) q n with (cid:101) q n ( z ) = z n q n (1 / ¯ z ).Denote by b n the Blaschke product q n / (cid:101) q n . It is easy to check that b n ( z ) b n (1 / ¯ z ) ≡ f − r n )( z ) = b n (1 / ¯ z )( l n,m A n,m /Q )( z ) , z ∈ O . where l n,m is the polynomial of degree m such that q n = q n,m l n,m . Choose (cid:15) > ⊂ D − (cid:15) (see Theorem 12). As l n,m is an arbitrary divisor of q n of degree m , wecan choose it to have zeros only in D − (cid:15) for all n large enough (this is possible since in fullproportion the zeros of q n approach K). Then it holds that(6.46) lim n →∞ | l n,m /Q | / n = 1uniformly on O . Further, by (3.2) and the last claim of Lemma 25, we have that(6.47) | f − r n | / n cap → exp (cid:26) − , T ) (cid:27) on T . ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 27
As any Blaschke product is unimodular on the unit circle, we deduce from (6.45)–(6.47)with the help of (6.37) ( i.e., A n,m goes to a constant) that | A n,m | / n cap → exp (cid:26) − , T ) (cid:27) in C \ K . Then we get from Lemma 22 that(6.48) lim sup n →∞ | A nm | / n ≤ exp (cid:26) − , T ) (cid:27) uniformly on closed subsets of C \ K, in particular, uniformly on O . Set q n,(cid:15) for the monicpolynomial whose zeros are those of q n lying in D − (cid:15) . Put n (cid:15) := deg( q n,(cid:15) ), (cid:101) q n,(cid:15) ( z ) = z n (cid:15) q n,(cid:15) (1 / ¯ z ), and let ν n,(cid:15) be the normalized counting measure of the zeros of q n,(cid:15) . As ν n ∗ → ω (K , T ) , it is easy to see that n (cid:15) /n → ν n,(cid:15) ∗ → ω (K , T ) when n → + ∞ . Thus,by the principle of descent (Section 7.2.5), it holds that(6.49) lim sup n →∞ | q n,(cid:15) | /n = lim sup n →∞ | q n,(cid:15) | /n (cid:15) ≤ exp {− V ω (K , T ) } , locally uniformly in C . In another connection, since log | − z ¯ u | is continuous for ( z, u ) ∈ D / (1 − (cid:15) ) × D − (cid:15) , it follows easily from the weak ∗ convergence of ν n,(cid:15) that(6.50) lim n →∞ | (cid:101) q n,(cid:15) ( z ) | /n = lim n →∞ | (cid:101) q n,(cid:15) ( z ) | /n (cid:15) = exp (cid:26)(cid:90) log | − z ¯ u | dω (K , T ) ( u ) (cid:27) , uniformly in D . Put b n,(cid:15) := q n,(cid:15) / (cid:101) q n.(cid:15) . Since the Green function of D with pole at u is givenby log | (1 − z ¯ u ) / ( z − u ) | , we deduce from (6.49), (6.50), and a simple majorization thatlim sup n →∞ | b n | /n ≤ lim sup n →∞ | b n,(cid:15) | /n ≤ exp (cid:8) − V ω (K , T ) D (cid:9) uniformly in D . Besides, the Green function of O is still given by log | (1 − z ¯ u ) / ( z − u ) | , hence V ω D (1 / ¯ z ) = V ω ∗ O ( z ), z ∈ O , where ω is any measure supported in D . Thus, we derive that(6.51) lim sup n →∞ | b n (1 / ¯ z ) | / n ≤ exp (cid:110) − V ω ∗ (K , T ) O ( z ) (cid:111) holds uniformly on O . Combining (6.45)–(6.51), we deduce thatlim sup n →∞ | f − r n | / n ≤ exp (cid:26) − , T ) − V ω ∗ (K , T ) O (cid:27) uniformly on O . This finishes the proof of the theorem since V ω ∗ (K , T ) C \ K = , T ) + V ω ∗ (K , T ) O in O by Lemma 25, the maximum principle for harmonic functions applied in O , and the factthat the difference of two Green potentials of the same measure but on different domains isharmonic in a neighborhood of the support of that measure by the first claim in Section 7.3. (cid:3) Proof of Corollary 8.
It follows from (3.3) and Lemma 25 thatlim sup n →∞ (cid:107) f − r n (cid:107) / n T ≤ exp (cid:26) − , T ) (cid:27) . On the other hand, by (3.2) and the very definition of convergence in capacity, we have forany (cid:15) > | f − r n | > (cid:18) exp (cid:26) − , T ) (cid:27) − (cid:15) (cid:19) n on T \ S n,(cid:15) , where cap( S n,(cid:15) ) → n → ∞ . In particular, it means that | S n,(cid:15) | → | S n,(cid:15) | is the arclength measure of S n,(cid:15) . Hence, we have thatlim inf n →∞ (cid:107) f − r n (cid:107) / n ≥ lim n →∞ (cid:18) | T \ S n,(cid:15) | π (cid:19) / n (cid:18) exp (cid:26) − , T ) (cid:27) − (cid:15) (cid:19) = exp (cid:26) − , T ) (cid:27) − (cid:15). As (cid:15) was arbitrary and since (cid:107) f − r n (cid:107) ≤ π (cid:107) f − r n (cid:107) T , this finishes the proof of the corollary. (cid:3) Proof of Theorem 6.
Let Θ be the conformal map of D onto G . Observe that Θ (cid:48) is a holo-morphic function in D with integrable trace on T since T is rectifiable [20, Thm. 3.12], andthat Θ extends in a continuous manner to T where it is absolutely continuous. Hence,( f ◦ Θ)(Θ (cid:48) ) / ∈ L . Moreover, g lies in E n ( G ) if and only if ( g ◦ Θ)(Θ (cid:48) ) / lies in H n := H M − n . Indeed, denote by E ∞ ( G ) the space of bounded holomorphic functionsin G and set E ∞ n ( G ) := E ∞ ( G ) M − n ( G ). It is clear that g ∈ E ∞ n ( G ) if and only if it ismeromorphic in G and bounded outside a compact subset thereof. This makes it obviousthat g ∈ E ∞ n ( G ) if and only if g ◦ Θ ∈ H ∞ n := H ∞ M − n , where H ∞ is the space of boundedholomorphic functions in D . It is also easy to see that E n ( G ) = E ( G ) E ∞ n ( G ). Since it isknown that g ∈ E ( G ) if and only if ( g ◦ Θ)(Θ (cid:48) ) / ∈ H [20, corollary to Thm. 10.1], theclaim follows. Notice also that g n is a best approximant for f from E n ( G ) if and only if( g n ◦ Θ)(Θ (cid:48) ) / is a best approximant for ( f ◦ Θ)(Θ (cid:48) ) / from H n . This is immediate fromthe change of variable formula, namely, (cid:107) f − g (cid:107) ,T = (cid:90) T | f ◦ Θ − g ◦ Θ | | Θ (cid:48) | dθ = (cid:107) ( f ◦ Θ)(Θ (cid:48) ) / − ( g n ◦ Θ)(Θ (cid:48) ) / (cid:107) , where we used the fact that | d Θ( e iθ ) | = | Θ (cid:48) ( e iθ ) | dθ a.e. on T [20, Thm. 3.11].Now, let g n be a best meromorphic approximants for f from E n ( G ). As L = H ⊕ ¯ H ,it holds that ( g n ◦ Θ)(Θ (cid:48) ) / = g + n + r n and ( f ◦ Θ)(Θ (cid:48) ) / = f + + f − , where g + n , f + ∈ H and r n , f − ∈ ¯ H . Moreover, it can be easily checked that r n ∈ R n and, as explained at thebeginning of the proof of Theorem 4, that f − ∈ A ( D ). Since by Parseval’s relation (cid:107) ( f ◦ Θ)(Θ (cid:48) ) / − ( g n ◦ Θ)(Θ (cid:48) ) / (cid:107) = (cid:107) f + − g + n (cid:107) + (cid:107) f − − r n (cid:107) , we immediately deduce that g + n = f + and that r n is an ¯ H -best rational approximant for f − .Moreover, by the conformal invariance of the condenser capacity (see (7.7)), cap(K , T ) =cap(Θ − (K) , T ). It is also easy to verify that K ∈ K f ( G ) if and only if Θ − ( K ) ∈ K f − ( D ).Hence, we deduce from Theorem 7 and the remark thereafter that | f − − r n | / n cap → exp (cid:26) V ω (Θ − , T ) D − − (K) , T ) (cid:27) in D \ Θ − (K) . The result then follows from the conformal invariance of the Green equilibrium measures,Green capacity, and Green potentials and the fact that, since Θ is locally Lipschitz-continuousin D , it cannot locally increase the capacity by more than a multiplicative constant [45, Thm.5.3.1]. (cid:3) Proof of Theorem 5.
By Theorem S and decomposition (7.16), the set K of minimal con-denser capacity for f is a smooth cut, hence a tame compact set with tame points ∪ γ j , suchthat ∂∂ n + V (cid:98) ω ( T, K) − ω ( T, K) = ∂∂ n − V (cid:98) ω ( T, K) − ω ( T, K) on (cid:91) γ j , ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 29 where (cid:98) ω ( T, K) is the balayage of ω ( T, K) onto K. As (cid:98) ω ( T, K) is the weighted equilibrium dis-tribution on K in the field V − ω ( T, K) (see (7.18)), the set K possesses the S-property in thesense of (4.4). If f is holomorphic in C \ K and since it extends continuously from both sideson each γ j with a jump that can vanish in at most countably many points, we get from [25,Thm. 1 (cid:48) ] that(6.52) lim n →∞ ρ / nn, ∞ ( f, T ) = exp (cid:26) − , T ) (cid:27) . However, Theorem 1 (cid:48) in [25] is obtained as an application of Theorem GR. Since the latteralso holds for functions in A ( G ), that is, those that are meromorphic in C \ K, (see theexplanation in the proof of Theorem 14), (6.52) is valid for these functions as well. As ρ n, ( f, T ) ≤ | T | ρ n, ∞ ( f, T ), where | T | is the arclength of T , we get from (6.52) thatlim sup n →∞ ρ / nn, ( f, T ) ≤ exp (cid:26) − , T ) (cid:27) . On the other hand, let g n be a best meromorphic approximants for f from E n ( G ) as inTheorem 6. Using the same notation, it was shown that (( f − g n ) ◦ Θ)(Θ (cid:48) ) / = ( f − − r n ),where r n is a best ¯ H -rational approximant for f − from R n . Hence, we deduce from thechain of equalities (cid:107) f − g n (cid:107) ,T = (cid:107) ( f ◦ Θ)(Θ (cid:48) ) / − ( g n ◦ Θ)(Θ (cid:48) ) / (cid:107) = (cid:107) f − − r n (cid:107) and Corollary 8 thatlim n →∞ (cid:107) f − g n (cid:107) / n ,T = exp (cid:26) − − (K) , T ) (cid:27) = exp (cid:26) − , T ) (cid:27) . As ρ n, ( f, T ) ≥ (cid:107) f − g n (cid:107) ,T by the very definition of g n and the inclusion R n ( G ) ⊂ E n ( G ),the lower bound for the limit inferior of ρ / nn, ( f, T ) follows. (cid:3) Some Potential Theory
Below we give a brief account of logarithmic potential theory that was used extensivelythroughout the paper. We refer the reader to the monographs [45, 47] for a thoroughtreatment.7.1.
Capacities.
In this section we introduce, logarithmic, weighted, and condenser capac-ities.7.1.1.
Logarithmic Capacity.
The logarithmic potential of a finite positive measure ω , com-pactly supported in C , is defined by V ω ( z ) := − (cid:90) log | z − u | dω ( u ) , z ∈ C . The function V ω is superharmonic with values in ( −∞ , + ∞ ] and is not identically + ∞ . The logarithmic energy of ω is defined by I [ ω ] := (cid:90) V ω ( z ) dω ( z ) = − (cid:90) (cid:90) log | z − u | dω ( u ) dω ( z ) . As V ω is bounded below on supp( ω ), it follows that I [ ω ] ∈ ( −∞ , + ∞ ].Let F ⊂ C be compact and Λ( F ) denote the set of all probability measures supportedon F . If the logarithmic energy of every measure in Λ( F ) is infinite, we say that F is polar .Otherwise, there exists a unique ω F ∈ Λ( F ) that minimizes the logarithmic energy over allmeasures in Λ( F ). This measure is called the equilibrium distribution on F and it is known that ω F is supported on the outer boundary of F , i.e., the boundary of the unboundedcomponent of the complement of F . Hence, if K and F are two compact sets with identicalouter boundaries, then ω K = ω F .The logarithmic capacity , or simply the capacity, of F is defined ascap( F ) = exp {− I [ ω F ] } . By definition, the capacity of an arbitrary subset of C is the supremum of the capacities ofits compact subsets. We agree that the capacity of a polar set is zero. It follows readilyfrom what precedes that the capacity of a compact set is equal to the capacity of its outerboundary.We say that a property holds quasi everywhere (q.e.) if it holds everywhere except on aset of zero capacity. We also say that a sequence of functions { h n } converges in capacity toa function h , h n cap → h , on a compact set K if for any (cid:15) > n →∞ cap ( { z ∈ K : | h n ( z ) − h ( z ) | ≥ (cid:15) } ) = 0 . Moreover, we say that the sequence { h n } converges in capacity to h in a domain D if itconverges in capacity on each compact subset of D . In the case of an unbounded domain, h n cap → h around infinity if h n (1 / · ) cap → h (1 / · ) around the origin.When the support of ω is unbounded, it is easier to consider V ω ∗ , the spherical logarithmicpotential of ω , i.e.,(7.1) V ω ∗ ( z ) = (cid:90) k ( z, u ) dω ( u ) , k ( z, u ) = − (cid:26) log | z − u | , if | u | ≤ , log | − z/u | , if | u | > . The advantages of dealing with the spherical logarithmic potential shall become apparentlater in this section.7.1.2.
Weighted Capacity.
Let F be a non-polar compact set and ψ be a lower semi-continuousfunction on F such that ψ < ∞ on a non-polar subset of F . For any measure ω ∈ Λ( F ), wedefine the weighted energy of ω by I ψ [ ω ] := I [ ω ] + 2 (cid:90) ψdω. Then there exists a unique measure ω F,ψ , the weighted equilibrium distribution on F , thatminimizes I ψ [ ω ] among all measures in Λ( F ) [47, Thm. I.1.3]. Clearly, ω F,ψ = ω F when ψ ≡ ω F,ψ admits the following characterization [47, Thm. I.3.3]. Let ω be apositive Borel measure with compact support and finite energy such that V ω + ψ is constantq.e. on supp( ω ) and at least as large as this constant q.e. on F . Then ω = ω F,ψ . The valueof V ω + ψ q.e. on supp( ω F,ψ ) is called the modified Robin constant and it can be expressedas(7.2) c ( ψ ; F ) = I ψ [ ω F,ψ ] − (cid:90) ψdω F,ψ = I [ ω F,ψ ] + (cid:90) ψdω
F,ψ . The weighted capacity of F is defined as cap ψ ( F ) = exp {− I ψ [ ω F,ψ ] } . Logarithmic energy with an external field is called weighted as it turns out to be an important objectin the study of weighted polynomial approximation [47, Ch. VI].
ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 31
Condenser Capacity.
Let now D be a domain with non-polar boundary and g D ( · , u )be the Green function for D with pole at u ∈ D . That is, the unique function such that(i) g D ( z, u ) is a positive harmonic function in D \ { u } , which is bounded outside eachneighborhood of u ;(ii) g D ( z, u ) + (cid:26) − log | z | , if u = ∞ , log | z − u | , if u (cid:54) = ∞ , is bounded near u ;(iii) lim z → ξ, z ∈ D g D ( z, u ) = 0 for quasi every ξ ∈ ∂D .For definiteness, we set g D ( z, u ) = 0 for any z ∈ C \ D , u ∈ D . Thus, g D ( z, u ) is definedthroughout the whole extended complex plane.It is known that g D ( z, u ) = g D ( u, z ), z, u ∈ D , and that the subset of ∂D for which (iii)holds does not depend on u . Points of continuity of g D ( · , u ) on ∂D are called regular , otherpoints on ∂D are called irregular; the latter form F σ polar set (in particular, it is totallydisconnected). When F is compact and non-polar, we define regular points of F as pointsof continuity of g D ( · , ∞ ), where D is the unbounded component of the complement of F . Inparticular, all the inner points of F are regular, i.e., the irregular points of F are containedin the outer boundary of F , that is, ∂D . We call F regular if all the point of F are regular.It is useful to notice that for a compact non-polar set F the uniqueness of the Greenfunction implies that(7.3) g C \ F ( z, ∞ ) ≡ − log cap( F ) − V ω F ( z ) , z ∈ C \ F, by property (ii) in the definition of the Green function and the characterization of theequilibrium potential (see explanation before (7.2)).In analogy to the logarithmic case, one can define the Green potential and the
Greenenergy of a positive measure ω supported in a domain D as V ωD ( z ) := (cid:90) g D ( z, u ) dω ( u ) and I D [ ω ] := (cid:90) (cid:90) g D ( z, w ) dω ( z ) dω ( w ) . Exactly as in the logarithmic case, if E is a non-polar compact subset of D , there exists aunique measure ω ( E,∂D ) ∈ Λ( E ) that minimizes the Green energy among all measures inΛ( E ). This measure is called the Green equilibrium distribution on E relative to D . The condenser capacity of E relative to D is defined ascap( E, ∂D ) := 1 /I D [ ω ( E,∂D ) ] . It is known that the Green potential of the Green equilibrium distribution satisfies(7.4) V ω ( E,∂D ) D ( z ) = 1cap( E, ∂D ) , for q.e. z ∈ E. Moreover, the equality in (7.4) holds at all the regular points of E . Furthermore, it is knownthat ω ( E,∂D ) is supported on the outer boundary of E . That is,(7.5) ω ( E,∂D ) = ω ( ∂ Ω ,∂D ) , where Ω is the unbounded component of the complement of E .Let F be a non-polar compact set, D any component of the complement of F , and E anon-polar subset of D . Then we define ω ( E,F ) and cap( E, F ) as ω ( E,∂D ) and cap( E, ∂D ),respectively. It is known that(7.6) cap(
E, F ) = cap(
F, E ) , where F and E are two disjoint compact sets with connected complements. That is, thecondenser capacity is symmetric with respect to its entries and only the outer boundary ofa compact plays a role in calculating the condenser capacity.As in the logarithmic case, the Green equilibrium measure can be characterized by theproperties of its potential. Namely, if ω has finite Green energy, supp( ω ) ⊆ E , V ωD is constantq.e. on supp( ω ) and is at least as large as this constant q.e. on E , then ω = ω ( E,∂D ) [47, Thm.II.5.12]. Using this characterization and the conformal invariance of the Green function, onecan see that the condenser capacity is also conformally invariant. In other words, it holdsthat(7.7) cap( E, ∂D ) = cap( φ ( E ) , ∂φ ( D )) , where φ is a conformal map of D onto its image.7.2. Balayage.
In this section we introduce the notion of balayage of a measure and describesome of its properties.7.2.1.
Harmonic Measure.
Let D be a domain with compact boundary ∂D of positive ca-pacity and { ω z } z ∈ D , be the harmonic measure for D . That is, { ω z } z ∈ D is the collection ofprobability Borel measures on ∂D such that for any bounded Borel function f on ∂D thefunction P D f ( z ) := (cid:90) f dω z , z ∈ D, is harmonic [45, Thm. 4.3.3] and lim z → x P D f ( z ) = f ( x ) for any regular point x ∈ ∂D atwhich f is continuous [45, Thm. 4.1.5].The generalized minimum principle [47, Thm. I.2.4] says that if u is superharmonic,bounded below, and lim inf z → x,z ∈ D u ( z ) ≥ m for q.e. x ∈ ∂D , then u > m in D unless u isa constant. This immediately, implies that(7.8) P D h = h for any h which is bounded and harmonic in D and extends continuously to q.e. point of ∂D .For z ∈ C and z (cid:54) = w ∈ D \ {∞} , set(7.9) h D ( z, w ) := (cid:26) log | z − w | + g D ( z, w ) , if D is bounded , log | z − w | + g D ( z, w ) − g D ( z, ∞ ) − g D ( w, ∞ ) , otherwise . Observe that by the properties of Green function h D ( z, · ) is harmonic at z . Moreover, itcan be computed using (7.3) that lim | zw |→∞ h D ( z, w ) = log cap( ∂D ) when D is unbounded.Therefore, h D ( z, w ) is defined for all w ∈ D and z ∈ C ∪ D . Moreover, for each w ∈ D , thefunction h D ( · , w ) is bounded and harmonic in D and extends continuously to every regularpoint of ∂D . It is also easy to see that h D ( z, w ) = h D ( w, z ) for z, w ∈ D . Hence, we deducefrom (7.8) that(7.10) h D ( z, w ) = (cid:26) P D (log | z − ·| )( w ) , if D is bounded ,P D (log | z − ·| − g ( z, ∞ ))( w ) , otherwise ,z ∈ ( C ∪ D ) \ ∂D , for w ∈ D and all regular w ∈ ∂D . ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 33
Balayage.
Let ν be a finite Borel measure supported in D . The balayage of ν , denotedby (cid:98) ν , is a Borel measure on ∂D defined by(7.11) (cid:98) ν ( B ) := (cid:90) ω t ( B ) dν ( t )for any Borel set B ⊂ ∂D . Since ω z ( ∂D ) = 1, the total mass of (cid:98) ν is equal to the total massof ν . Moreover, it follows immediately from (7.11) that (cid:98) δ z = ω z , z ∈ D . In particular, if D is unbounded, (cid:98) δ ∞ = ω ∞ = ω ∂D (for the last equality see [45, Thm. 4.3.14]). In other words, (cid:98) δ ∞ is the logarithmic equilibrium distribution on ∂D .It is a straightforward consequence of (7.11) that(7.12) (cid:90) f d (cid:98) ν = (cid:90) P D f dν for any bounded Borel function on ∂D . Thus, we can conclude from (7.8) and (7.12) that(7.13) (cid:90) hd (cid:98) ν = (cid:90) hdν for any function h which is bounded and harmonic in D and extends continuously to q.e.point of ∂D .Assume now that x ∈ ∂D is a regular point and W an open neighborhood of x in ∂D .Let further f ≥ ∂D which is supported in W and such that f ( x ) >
0. Since P D f ( z ) → f ( x ) when D (cid:51) z → x , we see from (7.12) that ˆ ν ( W ) >
0. Inparticular, ∂D \ supp(ˆ ν ) is polar.Let D (cid:48) be a domain with non-polar compact boundary such that D ⊂ D (cid:48) and let { ω (cid:48) z } z ∈ D (cid:48) be the harmonic measure for D (cid:48) . For any Borel set B ⊂ ∂D (cid:48) it holds that ω (cid:48) z ( B ) is a harmonicfunction in D with continuous boundary values on ∂D . Thus, (cid:90) ω (cid:48) z ( B ) d (cid:98) ν ( z ) = (cid:90) ω (cid:48) z ( B ) dν ( z )by (7.13). This immediately implies that(7.14) (cid:101) ν = (cid:101)(cid:98) ν, where (cid:101) ν is the balayage of ν onto ∂D (cid:48) . In other words, balayage can be done step by step.7.2.3. Balayage and Potentials.
It readily follows from (7.9), (7.10), and (7.13) that(7.15) (cid:90) h D ( z, w ) dν ( w ) = (cid:40) − V (cid:98) ν ( z ) , if D is bounded , − V (cid:98) ν ( z ) − g D ( z, ∞ ) , otherwise , z ∈ ( C ∪ D ) \ ∂D. Clearly, the left-hand side of (7.15) extends continuously to q.e. z ∈ ∂D . Thus, the sameis true for the right-hand side. In particular, this means that V (cid:98) ν is bounded on ∂D andcontinuous q.e. on ∂D . Hence, (cid:98) ν has finite energy.In the case when ν is compactly supported in D , formula (7.15) has even more usefulconsequences. Namely, it holds that(7.16) V νD ( z ) = V ν − (cid:98) ν ( z ) + c ( ν ; D ) , z ∈ C , where c ( ν ; D ) = (cid:82) g D ( z, ∞ ) dν ( z ) if D is unbounded and c ( ν ; D ) = 0 otherwise, and wherewe used a continuity argument to extend (7.16) to every z ∈ C . This, in turn, yields that(7.17) V (cid:98) ν ( z ) = V ν ( z ) + c ( ν ; D ) for q.e. z ∈ C \ D, where equality holds for all z ∈ C \ D and also at all regular points of ∂D . Moreover,employing the characterization of weighted equilibrium measures, we obtain from (7.17)that(7.18) (cid:98) ν = ω ∂D, − V ν and c ( − V ν ; ∂D ) = c ( ν ; D ) . If a measure ν is not compactly supported, the logarithmic potential of ν may not be de-fined. However, representations similar to (7.16)–(7.18) can be obtained using the sphericallogarithmic potentials. Indeed, it follows from (7.15) that( V ν ∗ − V (cid:98) ν − V νD )( z ) = (cid:90) [ k ( z, u ) + log | z − u | − g D ( u, ∞ )] dν ( u )= (cid:90) | u | > [log | u | − g D ( u, ∞ )] dν ( u ) − (cid:90) | u |≤ g D ( u, ∞ ) dν ( u ) . As the right-hand side of the chain of the equalities above is a finite constant and V νD vanishesquasi everywhere on ∂D , we deduce as in (7.16)–(7.18) that this constant is − c ( − V ν ∗ ; ∂D )and that(7.19) (cid:98) ν = ω ∂D, − V ν ∗ . Moreover, it holds that(7.20) V νD ( z ) = V ν ∗ ( z ) − V (cid:98) ν ( z ) + c ( − V ν ∗ ; ∂D ) , z ∈ C . Let now D be a bounded domain and K be a compact non-polar subset of D . If E ⊆ K is also non-polar and compact, then(7.21) | I D [ ω E ] − I [ ω E ] | ≤ max z ∈ K,u ∈ ∂D | log | z − u || by integrating both sides of (7.16) against ω E with ν = ω E . This, in particular, yields that(7.22) (cid:12)(cid:12)(cid:12)(cid:12) E, ∂D ) + log cap( E ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ max z ∈ K,u ∈ ∂D | log | z − u || . Weighted Capacity in the Field − U ν . Let ν be a probability Borel measure supportedin D , K ⊂ D r , r <
1, be a compact non-polar set, and D be the unbounded componentof the complement of K . Further, let U ν ( z ) = − (cid:82) log | − z ¯ u | dν ( u ) as defined in (4.1). Itis immediate to see that U ν = V ν ∗ ∗ , where, as usual, ν ∗ is the reflection of ν across T . Inparticular, it follows from (7.19), (7.20), and the characterization of the weighted equilibriumdistribution that(7.23) (cid:98) ν ∗ = ω K, − U ν , where (cid:98) ν ∗ is the balayage of ν ∗ onto ∂D relative to D . Thus, ω K, − U ν is supported on theouter boundary of K and remains the same for all sets whose outer boundaries coincide upto a polar set. In another connection, it holds that U ν ( z ) = − (cid:90) log | − z/u | dν ∗ ( u ) = − (cid:90) log | − z/u | d (cid:101) ν ∗ ( u ) = V (cid:101) ν ∗ ( z ) − V (cid:101) ν ∗ (0)for any z ∈ D r by (7.13) and harmonicity of log | − z/u | as a function of u ∈ D ∗ r , where (cid:101) ν ∗ is the balayage of ν ∗ onto T r . It is also true that (cid:98) ν ∗ = (cid:99)(cid:102) ν ∗ by (7.14). Thus, I ν [ K ] = I [ (cid:98) ν ∗ ] − (cid:90) V (cid:101) ν ∗ d (cid:98) ν ∗ + 2 V (cid:101) ν ∗ (0) , ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 35 where I ν [ K ] was defined in (4.2). Using the harmonicity of V (cid:98) ν ∗ + g D ( · , ∞ ) in D andcontinuity at regular points of ∂D , (7.13), the Fubini-Tonelli theorem, and (7.16), we obtainthat I ν [ K ] = (cid:90) (cid:16) V (cid:98) ν ∗ ( z ) + g D ( z, ∞ ) (cid:17) d (cid:98) ν ∗ ( z ) − (cid:90) V (cid:98) ν ∗ d (cid:101) ν ∗ + 2 V (cid:101) ν ∗ (0)= (cid:90) (cid:16) V (cid:101) ν ∗ D ( z ) − V (cid:101) ν ∗ ( z ) − c ( (cid:101) ν ∗ ; D ) + g D ( z, ∞ ) (cid:17) d (cid:101) ν ∗ ( z ) + 2 V (cid:101) ν ∗ (0)= I D [ (cid:101) ν ∗ ] − I [ (cid:101) ν ∗ ] + 2 V (cid:101) ν ∗ (0) . (7.24)Equation (7.24), in particular, means that the problem of maximizing I ν [ · ] among the setsin D r is equivalent to the problem of maximizing the Green energy of (cid:101) ν ∗ among the domainswith boundary in D r .7.2.5. Weak ∗ Convergence and Convergence in Capacity.
By a theorem of F. Riesz, thespace of complex continuous functions on C , endowed with the sup norm, has dual the spaceof complex measures on C normed with the mass of the total variation (the so-called strongtopology for measures). We say that a sequence of Borel measures { ω n } on C converges weak ∗ to a Borel measure ω if (cid:82) f dω n → (cid:82) f dω for any complex continuous function f on C . By the Banach-Alaoglu theorem, any bounded sequence of measures has a subsequencethat converges in the weak ∗ sense. Conversely, by the Banach-Steinhaus theorem, a weak ∗ converging sequence is bounded.We shall denote weak ∗ convergence by the symbol ∗ → . Weak ∗ convergence of measuresimplies some convergence properties of logarithmic and spherical logarithmic potentials,which we mention below.The following statement is known as the Principle of Descent [47, Thm. I.6.8]. Let { ω n } be a sequence of probability measures all having support in a fixed compact set. Supposethat ω n ∗ → ω and z n → z , z n , z ∈ C . Then V ω ( z ) ≤ lim inf n →∞ V ω n ( z n ) and I [ ω ] ≤ lim inf n →∞ I [ ω n ] . Weak ∗ convergence of measures entails some convergence in capacity of their spherical po-tentials. This is stated rather informally in [25, Sec. 3 and 4], but the result is slightly subtlebecause, as examples show, convergence in capacity generally occurs outside the support ofthe limiting measure only. A precise statement is as follows. Lemma 21.
Let { ω n } be a sequence of positive Borel measures such that ω n ∗ → ω . Then V ω n ∗ cap → V ω ∗ in C \ supp( ω ) . In particular, if ω is the zero measure, then the sphericallynormalized potentials V ω n ∗ converge to zero in capacity in the whole extended complex plane.Proof. Suppose first that ω n converges weak ∗ to the zero measure. Then the convergence isactually strong. Assume moreover that the measures ω n are supported on a fixed compactset K ⊂ C . Let G be a simply connected domain that contains K , L be a Jordan curvethat contains the closure of G in its interior, and D be a bounded simply connected domainthat contains L . Fix (cid:15) > E n := { z ∈ D : V ω n D ( z ) > (cid:15) } . By superharmonicityof V ω n D the set E n is open, and we can assume E n ⊂ G by taking n large enough. If E n isempty then cap( E n ) = 0, otherwise let E ⊂ E n be a nonpolar compact set. Then the Greenequilibrium potential V ω ( E,∂D ) D is bounded above by 1 / cap( E, ∂D ) [47, Thm. 5.11] which is In (4.2) we slightly changed the notation comparing to Section 7.1.2. Clearly, I ν [ · ] and cap ν ( · ) shouldbe I − U ν [ · ] and cap − U ν ( · ). Even though this change is slightly ambiguous, it greatly alleviates the notationthroughout the paper. finite. Hence h := V ω n D − ε cap( E, ∂D ) V ω ( E,∂D ) D is superharmonic and bounded below in in D \ E , with lim inf h ( z ) ≥ z tends to ∂E ∪ ∂D . By the minimum principle, we thushave V ω n D ≥ (cid:15) cap( E, ∂D ) V ω ( E,∂D ) D in D \ E. Set m := min u ∈ G min z ∈ L g D ( z, u ) > . Clearly, V ω ( E,∂D ) D ( z ) > m , z ∈ L , thus V ω n D ≥ (cid:15)m cap( E, ∂D ) on L. Hence, in view of (7.22) applied with K = G , we get − log cap( E n ) = − sup E ⊂ E n log cap( E ) ≥ (cid:15)m sup L V ω n D − C where C is independent of n . Using the uniform convergence to 0 of V ω n D on L , we get thatcap( E n ) → n → ∞ , that is, V ω n D cap → D . Let, as usual, (cid:98) ω n be the balayage of ω n onto ∂D . Since | (cid:98) ω n | = | ω n | → n → ∞ , we have that V (cid:98) ω n → D .Combining this fact with (7.16), we get that V ω n cap → D . Let u be an arbitrary point in G . Then { V ω n + | ω n | log |·− u |} is a sequence of harmonic functions in C \ G . It is easy to seethat this sequence converges uniformly to 0 there. As | ω n | log | · − u | cap → C , we deducethat V ω n cap → V ω n ∗ = V ω n + (cid:82) log + | u | dω n ( cf. (7.1)) since supp( ω n ) ⊂ K .Next, let { ω n } be an arbitrary sequence of positive measures that converges to the zeromeasure. As the restriction ω n | D converges to zero, we may assume by the first part of theproof that supp( ω n ) ⊂ O . It can be easily seen from the definition of the spherical potential(7.1) that(7.25) V ω n ∗ (1 /z ) = V ˜ ω n ∗ ( z ) + | ω n | log | z | , z ∈ C \ { } , where ˜ ω n is the reciprocal measure of ω n , i.e., ˜ ω n ( B ) = ω n ( { z : 1 /z ∈ B } ) for any Borel set B . Clearly ˜ ω n → ω n ) ⊂ D , thus from the first part of the proof we get V ˜ ω n ∗ cap → | ω n | →
0, we also see by inspection that | ω n | log | z | cap →
0. Therefore, by (7.25), weobtain that V ω n ∗ (1 /z ) cap → V ω n ∗ cap → { ω n } be a sequence of positive measures converging weak ∗ to some Borel measure ω (cid:54) = 0. If supp( ω ) = C , there is nothing to prove. Otherwise, to each ε >
0, we set F ε := { z ∈ C : d c ( z, supp( ω )) ≥ ε } where d c is the chordal distance on the Riemann sphere.Pick a continuous function f , with 0 ≤ f ≤
1, which is identically 1 on F ε and supported in F ε/ .By the positivity of ω n and its weak ∗ convergence to ω , we get0 ≤ lim n → + ∞ ω n ( F ε ) ≤ lim n → + ∞ (cid:90) f dω n = (cid:90) f dω = 0 . From this, it follows easily that if ε n → ω n := ω n | F εn converges strongly to the zero measure. Therefore V ω n ∗ cap → C by the previous part of theproof. Now, put ω n := ω n − ω n = ω n | C \ F εn . For fixed z ∈ C \ supp( ω ), the function k ( z, u )from (7.1) is continuous on a neighborhood of C \ F ε n for all n large enough. Redefining k ( z, u ) near z to make it continuous does not change its integral against ω nor ω n , therefore V ω ∗ ( z ) − V ω n ∗ ( z ) → n → + ∞ since ω n ∗ → ω . Moreover, it is straightforward to check ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 37 from the boundedness of | ω n | that the convergence is locally uniform with respect to z ∈ C .Finally, if supp( ω ) is bounded, we observe that when z → ∞ V ω ∗ ( z ) − V ω n ∗ ( z ) ∼ log | z | (cid:0) ω n ( C ) − ω ( C ) (cid:1) + (cid:90) log + | u | dω − (cid:90) log + | u | dω n which goes to zero in capacity since ω n ( C ) → ω ( C ) and log + is continuous in a neighborhoodof both supp( ω ) and supp( ω n ) for n large enough. This finishes the proof of the lemma. (cid:3) The following lemma is needed for the proof of Theorem 7.
Lemma 22.
Let D be a domain in C and { A n } be a sequence of holomorphic functions in D such that | A n | /n cap → c in D as n → ∞ for some constant c . Then lim sup n →∞ | A n | /n ≤ c uniformly on closed subsets of D .Proof. By the maximum principle, it is enough to consider only compact subsets of D andtherefore it is sufficient to consider closed disks. Let z ∈ D and x > D x := { w : | w − z | < x } is contained in D . We shall show that lim sup n →∞ (cid:107) A n (cid:107) /nD x ≤ c .Fix (cid:15) >
0. As | A n | /n cap → c on D x \ D x , there exists y n ∈ (2 x, x ) such that | A n | /n ≤ c + (cid:15) on L n := { w : | w − z | = y n } for all n large enough. Indeed, define S n := { w ∈ D x \ D x : || A n ( w ) | /n − c | > (cid:15) } . By the definition of convergence in capacity, we have thatcap( S n ) → n → ∞ . Further, define S (cid:48) n := {| w − z | : w ∈ S n } ⊂ [2 x, x ]. Since themapping w (cid:55)→ | w − z | is contractive, cap( S (cid:48) n ) ≤ cap( S n ) by [45, Thm. 5.3.1] and thereforecap( S (cid:48) n ) → n → ∞ . The latter a fortiori implies that | S (cid:48) n | → n → ∞ by [45, Thm.5.3.2(c)], where | S (cid:48) n | is the Lebesgue measure of S (cid:48) n . Thus, y n with the claimed propertiesalways exists for all n large enough. Using the Cauchy integral formula, we get that (cid:107) A n (cid:107) /nD x ≤ (cid:18) max L n | A n | πx (cid:19) /n ≤ c + (cid:15) n √ πx . As x is fixed and (cid:15) is arbitrary, the claim of the lemma follows. (cid:3) Green Potentials.
In this section, we prove some facts about Green potentials thatwe used throughout the paper. We start from the following useful fact.Let D and D be two domains with non-polar boundary and ω be a Borel measuresupported in D ∩ D . Then V ωD − V ωD is harmonic in D ∩ D . Clearly, this claim should be shown only on the support of ω . Using the conformalinvariance of Green potentials, it is only necessary to consider measures with compactsupport. Denote by (cid:98) ω and (cid:101) ω the balayages of ω onto ∂D and ∂D , respectively. Since V ωD = V ω − (cid:98) ω + c ( ω ; D ) and V ωD = V ω − (cid:101) ω + c ( ω ; D ) by (7.16) and V (cid:98) ω and V (cid:101) ω are harmonicon supp( ω ), it follows that V ωD − V ωD = V (cid:101) ω − (cid:98) ω + c ( ω ; D ) − c ( ω ; D ) is also harmonic there.7.3.1. Normal derivatives.
Throughout this section, ∂/∂ n i (resp. ∂/∂ n o ) will stand for thepartial derivative with respect to the inner (resp. outer) normal on the corresponding curve. Lemma 23.
Let L be a C -smooth Jordan curve in a domain D and V be a continuousfunction in D . If V is harmonic in D \ L , extends continuously to the zero function on ∂D and to C -smooth functions on each side of L , then V = − V σD , where σ is a signed Borelmeasure on L given by dσ = 12 π (cid:18) ∂V∂ n i + ∂V∂ n o (cid:19) ds and ds is the arclength differential on L . Proof.
As discussed just before this section, the distributional Laplacian of − V σD in D isequal to 2 πσ . Thus, according to Weyl’s Lemma and the fact that V = V σD ≡ ∂D , weonly need to show that ∆ V = 2 πσ . By the very definition of the distributional Laplacian,it holds that(7.26) (cid:90) (cid:90) D φ ∆ V dm = (cid:90) (cid:90) D V ∆ φdm = (cid:90) (cid:90) O V ∆ φdm + (cid:90) (cid:90) D \ O V ∆ φdm , for any infinitely smooth function φ compactly supported in D , where O is the interiordomain of L and dm is the area measure. According to Green’s formula (see (7.35) furtherbelow) it holds that(7.27) (cid:90) (cid:90) O V ∆ φdm = (cid:90) (cid:90) O ∆ V φdm + (cid:90) L (cid:18) φ ∂V∂ n i − V ∂φ∂ n i (cid:19) ds = (cid:90) L (cid:18) φ ∂V∂ n i − V ∂φ∂ n i (cid:19) ds as V is harmonic in O . Analogously, we get that(7.28) (cid:90) (cid:90) D \ O V ∆ φdm = (cid:90) L (cid:18) φ ∂V∂ n o − V ∂φ∂ n o (cid:19) ds, where we also used the fact that φ ≡ ∂D . Combining (7.27) and(7.28) with (7.26) and observing that ∂φ/∂ n i = − ∂φ/∂ n o yield (cid:90) (cid:90) D φ ∆ V dm = (cid:90) L (cid:18) ∂V∂ n i + ∂V∂ n o (cid:19) ds = 2 π (cid:90) L φdσ. That is, ∆ V = 2 πσ , which finishes the proof of the lemma. (cid:3) Lemma 24.
Let F be a regular compact set and G a simply connected neighborhood of F .Let also V be a continuous function in G that is harmonic in G \ F and is identically zeroon F . If L is an analytic Jordan curve in G such that V ≡ δ > on L , then π (cid:90) L ∂V∂ n i ds = − δ cap( F ∩ Ω , L ) , where Ω is the inner domain of L .Proof. It follows immediately from the maximum principle for harmonic functions, appliedin Ω \ F , that V = δ cap( F ∩ Ω , L ) V ωD in Ω, where D := C \ ( F ∩ Ω) and ω := ω ( L,F ∩ Ω) .Thus, it is sufficient to show that(7.29) 12 π (cid:90) L ∂V ωD ∂ n i ds = − . Observe that V ωD can be reflected harmonically across L by the assumption on V and there-fore normal inner derivative of V ωD does exist at each point of L . According to (7.16), itholds that(7.30) 12 π (cid:90) L ∂V ωD ∂ n i ds = 12 π (cid:90) L ∂V ω − (cid:98) ω ∂ n i ds, where (cid:98) ω is the balayage of ω onto F ∩ Ω. By Gauss’ theorem [47, Thm. II.1.1], it is truethat(7.31) 12 π (cid:90) L ∂V (cid:98) ω ∂ n i ds = (cid:98) ω (Ω) = (cid:98) ω ( F ∩ Ω) = 1 . ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 39
Since V ωD ≡ / cap( L, F ∩ Ω) outside of Ω and V (cid:98) ω is harmonic across L , we get from (7.31)and the analog of (7.30) with ∂ n i replaced by ∂ n , that(7.32) 12 π (cid:90) L ∂V ω ∂ n o ds = 12 π (cid:90) L ∂V (cid:98) ω ∂ n o ds = − π (cid:90) L ∂V (cid:98) ω ∂ n i ds = − . As ∂V ω /∂ n i and ∂V ω /∂ n o are smooth on L by (7.16), in particular, Lipschitz smooth, weobtain from [47, Thm. II.1.5] that dω = − π (cid:18) ∂V ω ∂ n i + ∂V ω ∂ n o (cid:19) ds and therefore(7.33) 12 π (cid:90) L ∂V ω ∂ n i ds = − ω ( L ) − π (cid:90) L ∂V ω ∂ n o ds = 0by (7.32). Finally, by plugging (7.31) and (7.33) into (7.30), we see the validity of (7.29).Hence, the lemma follows. (cid:3) Reflected sets.
In the course of the proof of Theorem 7, we used the conclusions ofLemma 25 below. It has to do with the specific geometry of the disk, and we could not findan appropriate reference for it in the literature.
Lemma 25.
Let E ⊂ D be a compact set of positive capacity with connected complement D , and E ∗ stand for the reflection of E across T . Further, let ω ∈ Λ( E ) be such that ω = (cid:98) ω ∗ , where ω ∗ is the reflection of ω across T and (cid:98) ω ∗ is the balayage of ω ∗ onto E . Then ω = ω ( E, T ) and (cid:101) ω = ω ( T ,E ) , where (cid:101) ω is the balayage of ω onto T relative to D . Moreover, itholds that V ω ∗ D = V (cid:101) ωD = 1 / cap( E, T ) − V ω D in D .Proof. Denote by (cid:101) ω and (cid:101) ω ∗ the balayage of ω onto T relative to D and the balayage of ω ∗ onto T relative to O . It holds that (cid:101) ω = (cid:101) ω ∗ . Indeed, since g O ( z, ∞ ) = log | z | , we get from(7.15) for z ∈ T that V (cid:101) ω ∗ ( z ) = (cid:90) [log | t | − log | z − t | ] dω ∗ ( t ) = (cid:90) [ − log | u | − log | z − / ¯ u | ] dω ( u )= − (cid:90) log | − z ¯ u | dω ( u ) = V ω ( z ) = V (cid:101) ω ( z ) , where we used the fact that z = 1 / ¯ z for z ∈ T and (7.17) applied to ω . Since both measures, (cid:101) ω and (cid:101) ω ∗ , have finite energy, the uniqueness theorem [47, Thm. II.4.6] yields that (cid:101) ω = (cid:101) ω ∗ .By (7.16), we have that V (cid:101) ωD = V (cid:101) ω − (cid:98)(cid:101) ω + c ( (cid:101) ω ; D ). Since (cid:98)(cid:101) ω = (cid:99)(cid:101) ω ∗ = (cid:98) ω ∗ = ω and by theequality (cid:101) ω = (cid:101) ω ∗ , (7.14), and the conditions of the lemma, it holds that(7.34) V (cid:101) ωD ( z ) = V (cid:101) ω − ω ( z ) + c ( (cid:101) ω ; D ) = c ( (cid:101) ω ; D ) − V ω D ( z ) , z ∈ C , where we used (7.16) once more. Hence, V ω D = c ( (cid:101) ω ; D ) q.e. on E and the unique char-acterization of the Green equilibrium distribution implies that ω = ω ( E, T ) and c ( (cid:101) ω ; D ) =1 / cap( E, T ). Moreover, it also holds that V (cid:101) ωD = c ( (cid:101) ω ; D ) = 1 / cap( E, T ) in O and therefore (cid:101) ω = ω ( T ,E ) , again by the characterization of the Green equilibrium distribution.The first part of the last statement of the lemma is independent of the geometry of thereflected sets and follows easily from (7.13) and the fact that for any z ∈ D the function g D ( z, u ) is a harmonic function of u ∈ O continuous on T . The second part was shown in(7.34). (cid:3) Dirichlet Integrals.
Let D be a domain with compact boundary comprised of finitelymany analytic arcs that possess tangents at the endpoints. In this section we only considerfunctions continuous on D whose weak ( i.e., distributional) Laplacian in D is a signedmeasure supported in D with total variation of finite Green energy, and whose gradient,which is smooth off the support of the Laplacian, extends continuously to ∂D except perhapsat the corners where its norm grows at most like the reciprocal of the square root of thedistance to the corner. These can be written as a sum of a Green potential of a signedmeasure as above and a harmonic function whose boundary behavior has the smoothnessjust prescribed above. By Proposition 11, the results apply for instance to V ω C \ Γ on C \ Γ assoon as ω has finite energy.Let u and v be two such functions. We define the Dirichlet integral of u and v in D by(7.35) D D ( u, v ) = − π (cid:90) (cid:90) D u ∆ vdm − π (cid:90) ∂D u ∂v∂ n ds, where ∆ v is the weak Laplacian of v and ∂/∂ n is the partial derivative with respect to theinner normal on ∂D . The Dirichlet integral is well-defined since the measure | ∆ v | has finiteGreen energy and is supported in D while | u∂v/∂ n | is integrable on ∂D . Moreover, it holdsthat(7.36) D D ( u, v ) = D D ( v, u ) . Indeed, this follows from Fubini’s theorem if u and v are both Green potentials and fromGreen’s formula when they are both harmonic. Thus, we only need to check (7.36) when v is harmonic and u is a Green potential. Clearly, then it should hold D D ( u, v ) = 0. Let a bea point in the support of ∆ u and ε > g D ( ., a ) which is so small thatthe open set A := { z ∈ D : g D ( z, a ) < ε } does not intersect the support of ∆ u . By ourchoice of ε , the boundary of A consists of ∂D and a finite union of closed smooth Jordancurves. Write v = v + v for some C ∞ -smooth functions v , v such that the support of v is included in A (hence v is identically zero in a neighborhood of D \ A where the closureis taken with respect to D ) while the support of v is compact in D . Such a decompositionis easily constructed using a smooth partition of unity subordinated to the open covering of D consisting of A and { z ∈ D : g D ( z, a ) > ε/ } . By the definition of the weak Laplacianwe have that D D ( v, u ) = − π (cid:90) (cid:90) D v ∆ udm − π (cid:90) ∂D v ∂u∂ n ds = − π (cid:90) (cid:90) D u ∆ v dm − π (cid:90) ∂D v ∂u∂ n ds = − π (cid:90) (cid:90) D u ∆ v dm − π (cid:90) (cid:90) D u ∆ v dm = 0 , where we used Green’s formula(7.37) (cid:90) (cid:90) A ( v ∆ u − u ∆ v ) dm = (cid:90) ∂A (cid:18) u ∂v ∂ n − v ∂u∂ n (cid:19) ds. Note that if γ ⊂ D is an analytic arc which is closed in D and u, v are harmonic across γ , then(7.38) D D ( u, v ) = D D \ γ ( u, v )because the rightmost integral in (7.35) vanishes on γ as the normal derivatives of v fromeach side of γ have opposite signs. ATIONAL APPROXIMANTS TO ALGEBRAIC FUNCTIONS 41
Observe also that if ν is a positive Borel measure supported in D with finite Green’senergy then ∆ V νD = − πν by Weyl’s lemma (see Section 7.3) and so by (7.35)(7.39) D D ( V νD ) := D D ( V νD , V νD ) = I D [ ν ] . Finally, if v is harmonic in D , it follows from the divergence theorem that(7.40) D D ( v ) = (cid:90) (cid:90) D (cid:107)∇ v (cid:107) dm , which is the usual definition for Dirichlet integrals. In particular, if D (cid:48) ⊂ D is a subdomainwith the same smoothness as D , and if we assume that supp ∆ v ⊂ D (cid:48) , we get from (7.38)and (7.40) that(7.41) D D ( v ) = D D (cid:48) ( v ) + D D \ D (cid:48) ( v ) = D D (cid:48) ( v ) + (cid:90) (cid:90) D \ D (cid:48) (cid:107)∇ v (cid:107) dm . Numerical Experiments
In order to numerically construct rational approximants, we first compute the truncatedFourier series of the approximated function (resulting rational functions are polynomialsin 1 /z that converge to the initial function in the Wiener norm) and then use Endymion software (it uses the same algorithm as the previous version
Hyperion [26]) to computecritical points of given degree n . The numerical procedure in Endymion is a descent algo-rithm followed by a quasi-Newton iteration that uses a compactification of the set R n whoseboundary consists of n copies of R n − and n ( n − / R n − [8]. This allows togenerate several initial conditions leading to a critical point. If the sampling of the bound-ary gets sufficiently refined, the best approximant will be attained. In practice, however,one cannot be absolutely sure the sampling was fine enough. This why we speak below ofrational approximants and do not claim they are best rational approximants. They are,however, irreducible critical points, up to numerical precision.In the numerical experiments below we approximate functions given by f ( z ) = 1 (cid:112) ( z − z )( z − z )( z − z )( z − z ) + 1 z − z , where z = 0 . . i , z = − . . i , z = − . . i , z = 0 . − . i , and z = − . − . i ;and f ( z ) = 1 (cid:112) ( z − z )( z − z )( z − z ) + 1 (cid:112) ( z − z )( z − z ) , where z = 0 . . i , z = − . . i , z = − . . i , z = − . − . i , and z = 0 . − . i .We take the branch of each function such that lim z →∞ zf j ( z ) = 2, j = 1 ,
2, and use first100 Fourier coefficients for each function.On the figures diamonds depict the branch points of f j , j = 1 ,
2, and disks denote thepoles of the corresponding approximants.
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