Multipoint Schur algorithm and orthogonal rational functions: convergence properties, I
aa r X i v : . [ m a t h . C A ] F e b MULTIPOINT SCHUR ALGORITHM AND ORTHOGONALRATIONAL FUNCTIONS: CONVERGENCE PROPERTIES.
L. BARATCHART, S. KUPIN, V. LUNOT, M. OLIVI
Abstract.
Classical Schur analysis is intimately connected to the the-ory of orthogonal polynomials on the circle [43]. We investigate here theconnection between multipoint Schur analysis and orthogonal rationalfunctions. Specifically, we study the convergence of the Wall rationalfunctions via the development of a rational analogue to the Szeg˝o the-ory, in the case where the interpolation points may accumulate on theunit circle. This leads us to generalize results from [22, 10], and yieldsasymptotics of a novel type.
Introduction
The theory of orthogonal polynomials, with respect to a positive mea-sure on the line or the circle, currently undergoes a period of intensivegrowth. To hint at recent advances, let us quote the papers by Killip-Simon[24], Mart´ınez-Finkelstein et al. [32], Mi˜na-D`ıaz [34], Kuijlaars et al. [27],McLaughlin-Miller, [31], Lubinsky [29] and Remling [41]. A comprehensiveaccount of many late developments in the field can be found in the mono-graph by Simon [43]. Let us mention in passing that, over the same period,non-Hermitian orthogonality with respect to complex measures, which isintimately connected with rational approximation and interpolation, madesome progress too; see, for example, Aptekarev [4], Aptekarev-Van Assche[5], Baratchart-K¨ustner-Totik [7] and Baratchart-Yattselev [8].The connection between orthogonal polynomials on the unit circle and theSchur algorithm is an old one. Recall that a Schur function is an analyticmap from the open unit disk into itself. The Schur algorithm, introduced bySchur and Nevanlinna [42, 35], associates to every Schur function a sequenceof complex numbers of modulus at most one, called its Schur (or Verblunsky)parameters. Since the mise en sc`ene of the present article unfolds mainly inthe framework of the Schur analysis, we shall call the parameters “Schur”,although the term “Verblunsky” seems to be more fair historically, see Si-mon [43, Sect. 1.1] for a discussion. These parameters may be viewed ashyperbolic analogues of the Taylor coefficients at the origin. They generate acontinued-fraction expansion of the function, whose truncations give rise tothe so-called Schur approximants. These are hyperbolic counterparts of the
Date : January, 25, 2010.1991
Mathematics Subject Classification.
Primary: 30B70. Secondary: 41A20.
Key words and phrases.
Approximation by rational functions, Schur algorithm, Schur(Verblunsky) parameters, orthogonal rational functions (orthogonal polynomials), Wallorthogonal functions (Wall polynomials).This work was partially supported by grants ANR-07-BLAN-024701 and ANR-09-BLAN-005801.
Taylor polynomials, see definition (0.2) to come. Now, an elementary linearfractional transformation puts Schur functions in one-to-one correspondencewith Carath´eodory functions, i.e. analytic functions with positive real partin the disk, which are themselves in bijection with positive measures on thecircle via the Herglotz transform. A long time ago already, Geronimus andWall observed the remarkable identity between the Schur parameters of afunction and the recurrence coefficients of the orthogonal polynomials associ-ated to the corresponding measure [19, 45]. However, only relatively recentlywas it stressed by Khrushchev [22, 23] how properties of the measure, thatgovern the convergence of the corresponding orthogonal polynomials, arelinked to the convergence of the Schur approximants on the unit circle.It must be pointed out that the Schur algorithm is among the seldomprocedures preserving the Schur character in rational approximation; equiv-alently, it yields Carath´eodory rational approximants to Carath´eodory func-tions on the disk or the half-plane. This feature is of fundamental im-portance in several areas of Physics and Engineering, where the Schur orCarath´eodory nature of a transfer function is to be interpreted as a passiv-ity property of the underlying system. Moreover, in such modeling issues,the relevant norms take place on the boundary of the analyticity domain,that is, on the circle or the line, see e.g. [33, 17, 3, 10]. This is why theresults by Khrushchev are of significance from the applied viewpoint as well,which was one incentive for the authors to undertake the present study. Thismotivation is illustrated in the doctoral work by V. Lunot [30].Unless the Schur function to be approximated possesses some symme-try, though, there is no particular reason why Schur approximants shoulddistinguish the origin. It is thus natural to turn to multipoint Schur ap-proximants, that play the role of Lagrange interpolating polynomials in thepresent hyperbolic context, see definitions (0.1) and (0.2) to come. The roleof orthogonal polynomials is then played by orthogonal rational functionswith poles at the reflections of the interpolation points across the unit cir-cle. Orthogonal rational functions, pioneered by Dzrbasjan [13], were laterstudied by Pan [38] and considerably expanded by Bultheel et al. [10], seealso Langer-Lasarow [28]. The last two references stress the connection withthe multipoint Schur algorithm, and the comprehensive exposition in [10],which contains further references, presents an account of Szeg˝o asymptoticswhen the interpolation points are compactly supported in the disk.The present article is concerned with the so-called determinate case (seecondition (0.3)) when the interpolation sequence may have limit points onthe circle, and its purpose is two-fold. On the one hand, we derive analoguesof Khrushchev’s results [22] on the convergence of Schur approximants inthe multipoint case, and on the other hand we present a counterpart ofthe Szeg˝o theory for the associated orthogonal rational functions. We limitourselves to regular measures on the circle, whose density does not vanish atlimit points of the interpolation sequence, and we do not touch upon what isperhaps the most important issue, namely how to choose the interpolationpoints in an optimal fashion as regards convergence rates. Nonetheless, thepresent paper seems first to propose asymptotics when the interpolationpoints approach the unit circle. ULTIPOINT SCHUR APPROXIMATION 3
Anyone writing on the subject faces the difficulty of expounding the mazeof formulas on which research can dwell. Our choice has been to give a tersesummary of what we use, along with references.0.1.
Definitions.
Let D be the open unit disk and T the unit circle. Afunction f is called Schur if it belongs to the unit ball of the Hardy space H ∞ ( D ), i.e. if f ∈ H ∞ ( D ) and || f || ∞ ≤
1. The collection of Schur functionsis called the
Schur class , indicated by S .The multipoint Schur algorithm goes as follows. Let ( α k ), for k ∈ N , bea fixed sequence of points in D . We set α = 0 by convention. Define theelementary factor ζ k by(0.1) ζ k ( z ) = z − α k − ¯ α k z , and put for f ∈ S , k ≥ f = f,γ k = f k ( α k +1 ) ,f k +1 = 1 ζ k +1 f k − γ k − ¯ γ k f k . We call f n the Schur remainder of f of order n . The Schur convergent ,or
Schur approximant to f of order n , is defined from (0.2) by formallycomputing f in terms of f n +1 and γ k for 0 ≤ k ≤ n , and then substituting f n +1 = 0 in the resulting expression.It is a straightforward consequence of the maximum principle, that thealgorithm stops at some finite n ( i.e. that f n is an unimodular constant) ifand only if f is a Blaschke product of degree n , namely a rational functionin S which is unimodular on T : B ( z ) = c n Y j =1 z − β j − ¯ β j z , where β j ∈ D , | c | = 1. Throughout the paper, we assume that this is not thecase , so that the Schur algorithm, when applied to f with some sequence( α k ), produces an infinite sequence ( f k ). By the maximum principle, it iseasily seen that f k is in turn Schur. The complex numbers γ k appearing inthe algorithm are called the Schur ( or Verblunsky) parameters of f , and ourassumption that f is not a finite Blaschke product is equivalent to the factthat γ k ∈ D for all k .The case where α k ≡
0, originally considered by Schur [42] and subse-quently studied by many authors, will be referred to as the classical
Schuralgorithm.
Thus, in the classical case, α k = 0 and ζ k ( z ) = z for all k , asopposed to the multipoint version above where ( α k ) may distribute arbitrarilyin D .It is clear from (0.2) (this is formalized in Proposition 1.2) that γ k iscompletely determined by the interpolation values f ( j ) ( α l ) with 0 ≤ j ≤ n l −
1, where n l is the multiplicity of α l in the sequence ( α ℓ ) ≤ ℓ ≤ k +1 andthe superscript ( j ) indicates the j -th derivative. In order for the Schurapproximants to actually converge to f , it is thus necessary that the sequence L. BARATCHART, S. KUPIN, V. LUNOT, M. OLIVI ( α k ) be a uniqueness set in H ∞ ( D ). This is equivalent to the negation ofBlaschke condition:(0.3) X k (1 − | α k | ) = + ∞ . Of importance to us will be the equivalence of (0.3) with the density ofrational functions having poles at the points (1 /α k ) in every Hardy space H p ( D ), 1 ≤ p < ∞ , as well as in the disk algebra A ( D ) [2, App. A].Next, we recall a basic construction relating the classical Schur algorithmto orthogonal polynomials on T , see e.g. [22, 43]. For µ a Borel probabilitymeasure on T , we let µ ac and µ s respectively be its absolutely continuousand singular components with respect to m , the Lebesgue measure given by dm ( t ) = dt/ (2 πit ) = π dθ where t = e iθ ∈ T . We further put µ ′ = dµ ac /dm so that dµ = µ ′ dm + dµ s .To f ∈ S , we associate two probability measures µ, ˜ µ on T by the relations(0.4) F µ ( z ) = 1 + zf − zf = Z T t + zt − z dµ ( t ) , F ˜ µ ( z ) = 1 − zf zf = Z T t + zt − z d ˜ µ ( t ) . Clearly F µ is a Carath´eodory function, i.e. Re F ( z ) > , z ∈ D ; moreover F (0) = 1. We call F µ the Herglotz transform of µ , and the representation(0.4) is possible because every Carath´eodory function is uniquely the Her-glotz transform of a finite positive measure. From the Fatou theorems [25,Ch. I, Sect. D], we note that(0.5) µ ′ = Re F µ = 1 − | f | | − zf | , lim r → Re F µ ( re iθ ) = + ∞ ,m -a.e. and µ s -a.e., respectively. Similar considerations hold for ˜ µ .Let ( φ n ) and ( ψ n ) be the orthonormal polynomials with respect to µ and˜ µ :(0.6) Z T φ n φ m dµ = δ nm , Z T ψ n ψ m d ˜ µ = δ nm , here δ nm is the Kronecker symbol. Our assumption that f is not a finiteBlaschke product means that µ and ˜ µ have infinite support, therefore φ n , ψ n have exact degree n . The sequences ( φ n ) and ( ψ n ) are called respectivelythe orthonormal polynomials of first and second kind associated with µ .Clearly φ n and ψ n are unique up to a multiplicative unimodular constant.We normalize them so that their respective leading coefficients k n and k ′ n are positive.For a polynomial φ of degree n , put φ ∗ ( z ) = z n φ (1 / ¯ z ). This is again apolynomial of degree n . Note that k n = φ ∗ n (0). The coefficients(0.7) ˜ γ n = ˜ γ n ( µ ) = − φ n +1 (0) k n +1 are called the Geronimus parameters associated with ( φ n ) (or with µ ).The following remarkable theorem, named after Geronimus, was provenalmost simultaneously by Geronimus [19] and Wall [45]. Theorem.
Let f ∈ S . If α k ≡ , the Schur parameters and the Geronimusparameters coincide, i.e. γ n = ˜ γ n , n ≥ . ULTIPOINT SCHUR APPROXIMATION 5
Since trading µ for ˜ µ is tantamount to change f into − f , a corollary isthat ˜ γ ( µ ) = − ˜ γ (˜ µ ).We turn to the multipoint version of Geronimus’ theorem, which is dueessentially to Bultheel et al. [10] although the first explicit statement isapparently in Langer-Lasarow [28]. For this, orthogonal polynomials needto be generalized into orthogonal rational functions whose construction wenow explain. Define the “partial” Blaschke products B k by(0.8) B ( z ) = 1 , B k ( z ) = B k − ( z ) ζ k ( z ) , where ζ k is given by (0.1) and k ≥
1. The functions {B , B , . . . , B n } spanthe space(0.9) L n = ( p n π n : π n ( z ) = n Y k =1 (1 − ¯ α k z ) , p n ∈ P n ) , where P n stands for the space of algebraic polynomials of degree at most n .In the classical case, that is when α k = 0 for all k , L n coincides with P n .Given a function g , we introduce the parahermitian conjugate g ∗ definedby g ∗ ( z ) = g (1 / ¯ z ). Observe that | g ∗ | = | g | on T and that ζ n ∗ = ζ n − , B k ∗ = B k − . For g ∈ L n , we set g ∗ = B n f ∗ ; clearly, g ∗ ∈ L n . There is nonotational discrepancy since in the classical case the star operation agreeswith the definition we gave before. Put B n,i = Q nk = i ζ k . Each g ∈ L n can beuniquely decomposed in the form g = a n B n + a n − B n − + · · · + a B + a , and then g ∗ = ¯ a B n, + ¯ a B n, + · · · + ¯ a n − B n,n − + ¯ a n − B n,n + ¯ a n . It is plain that a n = g ∗ ( α n ) and a = g ( α ).Now, pick a Schur function f which is not a Blaschke product, denote itsHerglotz measure by µ (0.4), and consider L n as a subspace of L ( µ ). This ispossible since µ has infinite support. Let ( φ k ) ≤ k ≤ n be an orthonormal basisfor L n such that φ = 1 and φ k ∈ L k \ L k − . Such a basis is easily obtainedon applying the Gram-Schmidt orthonormalization process to B , B , . . . , B n .We customary write(0.10) φ n = κ n B n + a n,n − B n − + . . . + a n, B + a n, B , where κ n = φ ∗ n ( α n ). Definition 0.1.
The functions ( φ k ) are called the orthogonal rational func-tions of the first kind associated to ( α k ) and µ . The ( ψ n ) arising from embedding L n to L (˜ µ ) are called the orthogo-nal rational functions of the second kind. Clearly, the orthogonal rationalfunctions ( φ n ) , ( ψ n ) defined in (0.10) reduce to the orthonormal polynomialsfrom (0.6) in the classical case.Generically, the dependence on the nodes ( α k ) and the measure µ will beomitted. The words “orthogonal rational function” will be abbreviated asORF or OR-function. L. BARATCHART, S. KUPIN, V. LUNOT, M. OLIVI
The definition of the
Geronimus parameters (˜ γ k ) for OR-functions is˜ γ n = − φ n ( α n − ) φ ∗ n ( α n − ) , n ≥ . Note we do not define ˜ γ and there is a shift of index as compared to (0.7).It is quite nontrivial that one can relate the Schur algorithm (0.2) andthe ORFs (0.10) in the multipoint case as well: Theorem ([10, 28]) . Let ( α k ) , f ∈ S , and the ORFs ( φ n ) be as above. Thenthe multipoint Schur and Geronimus parameters coincide, i.e. γ k = ˜ γ k +1 . We prove this fundamental result in Section 2 for the sake of completeness.0.2.
Discussion of the main results.
The convergence properties of theSchur approximants and of the ORFs ( φ n ) are the main address of thepresent work, which is in part inspired by the results obtained by Khrushchev[22]. To better see the parallel between the classical and the multipointcase, we give below a sample of results from [22] in the classical situation,and have them followed by their multipoint counterparts, numbered with aprime superscript; we connect these counterparts to the forthcoming resultsin between parentheses.We say that a measure µ is Erd˝os, iff µ ′ > T . This is equivalentto say that | f | < T . Theorem 1 ([22], Theorem 1) . Let f ∈ S and µ be its Herglotz measure. If α k ≡ , then µ is Erd˝os if and only if the Schur remainders f n satisfy lim n Z T | f n | dm = 0 . The next result is stated in terms of the classical Wall polynomials A n , B n of f [22, Sect. 4], obtained from Definition 1.5 below by setting α k ≡
0. Bydefinition of the Wall polynomials, the ratio A n /B n is the Schur approximant to f of degree n . Recall that the pseudohyperbolic distance on D is definedas ρ ( z, w ) = | z − w | / | − ¯ wz | , z, w ∈ D . Theorem 2 ([22], Corollary 2.4) . A measure µ is Erd˝os if and only if lim n Z T ρ (cid:18) f, A n B n (cid:19) dm = 0 . We shall see that, in the multipoint situation when the sequence ( α k )accumulates on the unit circle, the conclusions of Theorems 1 and 2 getlocalized around the accumulation points of ( α k ) on T so that L -norms getweighted by the Poisson kernel at α n +1 . This is why, somewhat reminiscentlyof the Fatou theorem, we put extra-conditions on µ , locally around suchpoints, to derive convergence properties. Namely, let Acc ( α k ) = ( α k ) \ ( α k )be the set of accumulation points of ( α k ); the bar (or clos ( . )) stands for theclosure of a set. The following assumptions play an important role in ourproofs µ ′ ∈ C ( O ( Acc ( α k ) ∩ T )) , (0.11) µ ′ > O ( Acc ( α k ) ∩ T ) , (0.12) { Acc ( α k ) ∩ T } ⊂ T \ supp µ s , (0.13) ULTIPOINT SCHUR APPROXIMATION 7 where, for A ⊂ T , O ( A ) designates an open neighborhood of A in T and C ( A ) is the space of continuous functions on A . The closed support of µ s isdenoted by supp µ s . When the sequence ( α k ) accumulates nontangentiallyon Acc ( α k ) ∩ T , meaning that every convergent subsequence to ξ ∈ T tendsto the latter nontangentially, two weaker substitutes for (0.11), (0.12) arealso of interest:each ξ ∈ Acc ( α k ) ∩ T is a Lebesgue point of µ ′ , p µ ′ , (0.14) and µ ′ ( ξ ) > µ ′ is upper semicontinuous , < δ < µ ′ < M < ∞ on(0.15) O ( Acc ( α k ) ∩ T ) , and each ξ ∈ Acc ( α k ) ∩ T is aLebesgue point of log µ ′ . From (0.5), we see that (0.11) and (0.12) may be ascertained in terms of f ,namely f ∈ C ( O ( Acc ( α k ) ∩ T )) and | f | < Acc ( α k ) ∩ T ) ⊂ T \ clos { z : zf ( z ) = 1 } .The multipoint analogues of the previous theorems go as follows. Theorem 1’ (Corollary 3.5 and Theorem 4.6) . Let (0.3) , (0.11) - (0.13) hold,and | f | < a.e. on T . Then lim k Z | f k | P ( ., α k ) dm = 0 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then one can replacehypotheses (0.11) and (0.12) with (0.14) . Above, P ( ., α k ) is the Poisson kernel at α k on D (0.18). Denote by W q,p ( T )the Sobolev spaces on T (see Section 0.3 for more details). Recall that( A n ) , ( B n ) are the Wall rational functions from Definition 1.5 correspondingto f ∈ S . Theorem 1”.
Let µ be absolutely continuous with µ ′ ∈ W − /p,p ( T ) forsome p > , and µ ′ > on some neighborhood O ( Acc ( α k ) ∩ T ) . Then lim n (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) f − A n B n (cid:19) p P ( ., α n +1 ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ = 0 . Remarks on the converse to Theorem 1’ follow Theorems 3.2, 3.4.
Theorem 2’ (Theorem 4.2) . Assumptions being as in Theorem 1’, we have lim k Z T ρ (cid:18) f, A k B k (cid:19) P ( ., α k +1 ) dm = 0 . The final point of the paper is to carry over the Szeg˝o theory to themultipoint setting. Recall that a measure µ is called Szeg˝o (notation: µ ∈ (S)) iff log µ ′ ∈ L ( T ). For µ ∈ (S), the associated Szeg˝o function S is(0.16) S ( z ) = S [ µ ]( z ) := exp (cid:18) Z T t + zt − z log µ ′ dm ( t ) (cid:19) . The function S given by (0.16) is the so-called outer function in H ( D ) suchthat | S | = µ ′ a.e. on T , normalized so that S (0) > L. BARATCHART, S. KUPIN, V. LUNOT, M. OLIVI
The first version of the next theorem, which addresses the classical case,was proven by Szeg˝o [44]. Subsequent improvement were obtained by Geron-imus [20], Krein [26], and others; see Simon [43] for the discussion and a fulllist of references. Some of the latest improvements are due to Nikishin-Sorokin [36], Peherstorfer-Yuditskii [39]. A generalized version of Szeg˝ocondition is treated in Denisov-Kupin [14, 15].
Theorem 3.
Let µ ∈ (S) and ( φ n ) be the corresponding orthonormal poly-nomials (0.6) . Then • lim n ( Sφ ∗ n )(0) = 1 ; more generally, lim n ( Sφ ∗ n )( z ) = 1 for z ∈ D . • lim n Z T | Sφ ∗ n − | dm = 0 . Moreover, µ ∈ (S) if and only if lim n Z T P (cid:18) f, A n B n (cid:19) dm = 0 , where P ( ., . ) is the hyperbolic distance on D (4.1) . Equivalently, lim n Z T log(1 − | f n | ) dm = 0 . The last assertion of the theorem concerning the hyperbolic distance isfrom Khrushchev [22], Theorem 2.6.A multipoint analogue to the previous theorem when ( α n ) is compactlysupported in D is Theorem 9.6.9 from Bultheel et al. [10]; its generalizationto sequences ( α k ) meeting (0.3) is given below. It is more difficult andrequires some preparation. It relies on a priori pointwise estimates of ( φ n )(see Proposition 5.6), that play here the role of classical bounds by Szeg˝oand Geronimus [44, Ch. 12], [20, Ch. 4]. Such estimates are new even inthe polynomial case, as they handle some situations where µ ′ may vanish.Their proof in turn depends on ∂ -estimates and Sobolev embeddings. Ascompared to the case where ( α n ) is compactly supported in D , the resultbelow is of new type in that Sφ ∗ n is asymptotic to a normalized Cauchy kernelat the last interpolation point, which is unbounded when ( α n ) approaches T . Here is a combination of Theorem 4.3, Theorem 5.8 and Corollary 5.13to come: Theorem 3’.
Let (0.3) , (0.11) - (0.13) be in force, with µ ∈ (S) . Then • lim n | φ ∗ n ( α n ) | | S ( α n ) | (1 − | α n | ) = 1 ; more generally, for any se-quence ( z n ) ⊂ D , it holds lim n ( φ ∗ n ( z n ) S ( z n ) p − | z n | − β n p − | α n | p − | z n | − α n z n ) = 0 , where β n = ( Sφ ∗ n )( α n ) / | ( Sφ ∗ n )( α n ) | . In particular, for a fixed z ∈ D , lim n ( Sφ ∗ n ( z ) − β n p − | α n | − α n z ) = 0 . • We also have lim n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Sφ ∗ n ( z ) − β n p − | α n | − α n z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 . ULTIPOINT SCHUR APPROXIMATION 9 • Moreover lim n Z T P (cid:18) f, A n B n (cid:19) P ( ., α n +1 ) dm = 0 , in particular lim n Z T log(1 − | f n | ) P ( ., α n ) dm = 0 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then one can replace (0.11) and (0.12) by (0.15) . It would be interesting to know how much these assumptions can berelaxed. In particular, we shall give an example where the conclusion ofTheorem 3’ holds although (0.12) fails.The paper is organized as follows. The multipoint Schur algorithm, itsconnections to continued fractions, the Wall rational functions and the Schurparameters are discussed in Section 1. Section 2 introduces the ORFs( φ n ) , ( ψ n ) , and expresses them through Geronimus parameters and transfermatrices. The construction is then used to prove Geronimus’ theorem andits corollaries. Although we use a different normalization, by and large, thecontent of Section 2 is borrowed from Bultheel et al. [10]. The convergenceof Schur remainders and Wall RFs is studied in Sections 3 and 4. Section 5is devoted to the discussion of the Szeg˝o-type theorem and its corollaries.0.3. Some notation.
As already mentioned, the closure of A ⊂ C is in-dicated by clos A or A , while O ( A ) designates an open neighborhood of A in T . The normalized Lebesgue measure on T is denoted by m , and themeasure of A ⊂ T is denoted by | A | . We put C ( A ) for the space of con-tinuous functions on A . The symbol || . || p stands for the usual norm on theLebesgue space L p ( T ) , ≤ p ≤ ∞ ; when p = 2 , the subindex is usuallydropped . The classical analytic Hardy spaces of the disk are denoted by H p ( D ) , ≤ p ≤ ∞ , and A ( D ) is the disk algebra, comprised of analyticfunctions in D that extend continuously to D , endowed with the sup norm.Standard references on the subject are the books by Duren [16], Garnett[18], Koosis [25], from which we often quote basic facts without further ci-tation. In particular, H p -functions have well-defined nontangential limits in L p ( T ), and we use the same notation for the function in D and its trace on T . Every real-valued ϕ ∈ L ( T ) is m -a.e. the real part of the nontangentiallimit of the complex analytic function(0.17) F ϕ ( z ) = Z T t + zt − z ϕ ( t ) dt, z ∈ D , which is called the Herglotz transform of ϕ . The map sending ϕ to the imag-inary part of F ϕ is the conjugation operator , denoted with the superscript“ˇ”, i.e. F ϕ = ϕ + i ˇ ϕ ; it extends linearly to complex-valued functions. Bya theorem of M. Riesz, the conjugation operator acts on L p ( T ) , < p < ∞ .Moreover, since it is of convolution type, it commutes with d/ | dt | and inte-grating by parts one sees that it also acts on W ,p ( T ), the space of absolutelycontinuous functions with L p derivative on T . The Poisson kernel on D is(0.18) P ( z, w ) = P w ( z ) = (1 − | w | ) / | z − w | , where z ∈ T , w ∈ D .We shall need some basic facts from Sobolev space theory for which werefer the reader to Adams-Fournier [1]. In particular, for I ⊂ T an open arc,those ϕ for which(0.19) Z t,t ′ ∈ I (cid:12)(cid:12)(cid:12)(cid:12) ϕ ( t ) − ϕ ( t ′ ) t − t ′ (cid:12)(cid:12)(cid:12)(cid:12) p dm ( t ) dm ( t ′ ) < ∞ , with 1 < p < ∞ , form the fractional Sobolev space W − /p,p ( I ) (that co-incides with the Besov space B − /p,pp ( I )). It is a real interpolation spacebetween W ,p ( I ) and L p ( I ): W − /p,p ( I ) = [ W ,p ( I ) , L p ( I )] /p , that embedscompactly into L p ( I ). By interpolation [1, Sect. 7.3.2, Theorem 7.3.2], theconjugation operator also acts on W − /p,p ( T ).When h is defined on E ⊂ C and 0 < α <
1, we say that h is H¨oldercontinuous of exponent α if there is a constant C > | h ( t ) − h ( t ′ ) | ≤ C | t − t ′ | α for all t, t ′ ∈ E . We then write h ∈ H α ( E ). Notethat H α ( I ) ⊂ W − /p,p ( I ) if 1 < p < / (1 − α ). By the Sobolev embeddingtheorem [1, Theorem 4.12], it holds conversely that W − /p,p ( I ) ⊂ H − /p ( I )for p >
2. 1.
Wall rational functions
In this section we rewrite Section 4 from [22] for the multipoint case. Thepresentation is very close to the original, and only some technical details aredifferent. Roughly speaking, one mainly has to replace z n with B n ; that iswhy we generically give results accompanied by precise references to [22] andomit the proofs. We start recalling basic definitions on continued fractions[46].A continued fraction is an infinite expression of the form b + a b + a b + a b a ... . We conform the more economic notation b + a b + a b + a b + . . . . For any complex-valued ω , we let t ( ω ) = b + ω and, for k ≥ t k ( ω ) = a k b k + ω . By definition, the n -th convergent P n /Q n of the continued fraction is P n Q n = t ◦ t ◦ · · · ◦ t n (0) = b + a b + a b + . . . + a n b n . ULTIPOINT SCHUR APPROXIMATION 11
Proposition 1.1 ([22], relations (3.2)-(3.4)) . The quantities P n and Q n canbe computed according to the recurrence relations: P − = 1 , Q − = 0 ,P = b , Q = 1 ,P k +1 = b k +1 P k + a k +1 P k − Q k +1 = b k +1 Q k + a k +1 Q k − for k ≥ . More generally, P n − ω + P n Q n − ω + Q n = t ◦ t ◦ · · · ◦ t n ( ω ) . First we record the following fact.
Proposition 1.2.
For k ≥ , γ k depends only on f ( i ) ( α j ) , ≤ j ≤ k + 1 , ≤ i < m j , where m j is the multiplicity of α j at the k -th step, i.e. m j isthe number of times the value of α j enters ( α l ) ≤ l ≤ k +1 .Proof. Noticing in case of repetitions that f j ( α j ) = f ′ j − ( α j ) −| α j | −| f j − ( α j ) | ,the proof is immediate by induction on (0.2). (cid:3) We now rewrite the recursive step of (0.2) as(1.1) f k − = γ k − + (1 − | γ k − | ) ζ k ¯ γ k − ζ k + f k . For ω ∈ D \ { } , set(1.2) τ k ( ω ) = τ k ( ω, z ) := γ k + (1 − | γ k | ) ζ k +1 ¯ γ k ζ k +1 + ω , and put τ k (0) = γ k . Hence, f k = τ k ( f k +1 ) and(1.3) f = τ ◦ τ ◦ · · · ◦ τ n ( f n +1 ) . In a way reminiscent of how we defined P n /Q n , we obtain the Schur conver-gent R n of degree n upon replacing f n +1 by 0 in (1.3), that is,(1.4) R n = τ ◦ τ ◦ · · · ◦ τ n − ◦ τ n (0) = τ ◦ τ ◦ · · · ◦ τ n − ( γ n ) . Proposition 1.3.
The rational function R n interpolates f at ( α k ) ≤ k ≤ n +1 ,counting multiplicities, and their first n + 1 Schur parameters coincide.Proof.
Note that τ k ( ω, α k +1 ) = γ k is independent of ω . Thus, for 0 ≤ k ≤ n , f ( α k +1 ) = τ ◦ · · · ◦ τ k ( τ k +1 ◦ · · · ◦ τ n ( f n +1 ) , α k +1 )= τ ◦ · · · ◦ τ k ( τ k +1 ◦ · · · ◦ τ n (0) , α k +1 )= R n ( α k +1 ) . Consequently, R n interpolates f at the point α k +1 .The remaining part of the claim is proven by induction. The base ofinduction being obvious, suppose that the k first Schur parameters of f and R n coincide. Then, denoting R [1] n , . . . R [ n ] n the Schur remainders of R n , wesee that R [ k ] n = τ − k − ◦ · · · ◦ τ − ( R n ), and R [ k ] n ( α k +1 ) = τ − k − ◦ · · · ◦ τ − ( R n , α k +1 ) = τ − k − ◦ · · · ◦ τ − ◦ τ ◦ τ ◦ · · · ◦ τ n − ( γ n , α k +1 )= τ k ( τ k +1 ◦ · · · ◦ τ n − ( γ n ) , α k +1 ) = γ k +1 . Therefore, the k + 1-th Schur parameter of R n and f coincide. (cid:3) The Schur algorithm can be readily connected to the continued fractions.Indeed, let P n /Q n be the sequence of convergents associated to(1.5) γ + (1 − | γ | ) ζ ¯ γ ζ + 1 γ + (1 − | γ | ) ζ ¯ γ ζ + . . . . Then, the functions R n are none but the P n /Q n .For n ≥
1, we have by Proposition 1.1 P n = γ n P n − + P n − Q n = γ n Q n − + Q n − P n − = ¯ γ n − ζ n P n − + (1 − | γ n − | ) ζ n P n − Q n − = ¯ γ n − ζ n Q n − + (1 − | γ n − | ) ζ n Q n − (1.6)with P − = 1 , P = γ , Q − = 0 , Q = 1 . For P n and Q n , we easily prove the next lemma. Lemma 1.4 ([22], Lemma 4.1) . For n ≥ , we have P n +1 , Q n +1 ∈ L n +1 , P n , Q n ∈ L n and P n +1 = ζ n +1 Q ∗ n , Q n +1 = ζ n +1 P ∗ n . Mimicking [22], formulas (4.5), (4.12), we get(1.7) (cid:20) Q ∗ n P ∗ n P n Q n (cid:21) = Y k = n (cid:20) γ k γ k (cid:21) (cid:20) ζ k
00 1 (cid:21)! (cid:20) γ γ (cid:21) , with n ≥
1. Let us set A n = P n , B n = Q n and choose the representative R n = A n /B n for R n . Definition 1.5. A n and B n are called the n -th W all rational functionsassociated to the Schur function f and the sequence ( α k ) . In the new notation, the previous relation reads as
Proposition 1.6 ([22], relation (4.12)) . We have (cid:20) B ∗ n A ∗ n A n B n (cid:21) = Y k = n (cid:20) γ k γ k (cid:21) (cid:20) ζ k
00 1 (cid:21)! (cid:20) γ γ (cid:21) . (1.8)The dependence of A n , B n on f and ( α k ) will be usually dropped. Forconvenience, we abbreviate “Wall rational function” as WRF or Wall RF. Corollary 1.7 ([22], relations (4.14), (4.15)) . The Wall RFs A n , B n havethe following properties: (1) B n ( z ) B ∗ n ( z ) − A n ( z ) A ∗ n ( z ) = B n ( z ) ω n , (2) | B n ( ξ ) | − | A n ( ξ ) | = ω n on T , (3) f ( α i ) = A n /B n ( α i ) = B ∗ n /A ∗ n ( α i ) , for ≤ i ≤ n + 1 , ULTIPOINT SCHUR APPROXIMATION 13 where ω n = n Y k =0 (1 − | γ k | ) . The proof is by taking the determinant in (1.8).
Proposition 1.8 ([22], Lemma 4.5) . For n ≥ , we have (1) A n , A ∗ n and B n lie in L n , (2) B n does not vanish on D , (3) A n /B n and A ∗ n /B n are Schur functions. These preparations bring us to the following important
Theorem 1.9 ([22], Theorem 4.6) . The Wall RFs A n and B n are connectedto f and f n +1 by (1.9) f ( z ) = A n ( z ) + ζ n +1 ( z ) B ∗ n ( z ) f n +1 ( z ) B n ( z ) + ζ n +1 ( z ) A ∗ n ( z ) f n +1 ( z ) . The theorem shows that, in Nevanlinna’s parametrization of all Schurinterpolants to f at ( α k ) ≤ k ≤ n +1 [18, Ch. IV, Lemma 6.1], the value zerofor the parameter yields R n = A n /B n while the value f n +1 yields f .2. ORFs and Geronimus’ theorem
The results of this section are borrowed from [10, 11]. We formulate theresults and briefly discuss them for the completeness of presentation; theproofs are generically omitted.2.1.
Orthogonal rational functions.
Let µ be a positive probability mea-sure on T with infinite support. Obviously, L n is a (closed) subspace of L ( µ ), and, following [10, Ch. 3], we regard it as a reproducing kernelHilbert space. The reproducing kernels for L n are easily seen to satisfy theso-called Christoffel-Darboux relations, which can be interpreted as recur-rence relations for the ORFs ( φ n ). Namely, we have Theorem 2.1 ([10], Theorem 4.1.1) . For n ≥ , it holds that (cid:20) φ n ( z ) φ ∗ n ( z ) (cid:21) = T n ( z ) (cid:20) φ n − ( z ) φ ∗ n − ( z ) (cid:21) , where T n ( z ) = s − | α n | − | α n − | p − | ˜ γ n | − ¯ α n − z − ¯ α n z (cid:20) − ˜ γ n − ˜ γ n (cid:21) (2.1) × (cid:20) λ n
00 ¯ λ n (cid:21) (cid:20) ζ n − ( z ) 00 1 (cid:21) , and ˜ γ n = − φ n ( α n − ) φ ∗ n ( α n − ) , η n = 1 − α n ¯ α n − − ¯ α n α n − , (2.2) λ n = | − ¯ α n α n − | − α n ¯ α n − φ ∗ n ( α n − ) | φ ∗ n ( α n − ) | κ n − | κ n − | η n . (2.3) Definition 2.2.
We call ˜ γ n , given by (2.2) , the n -th Geronimus parameterof the measure µ (with respect to the sequence ( α k ) ). Corollary 3.1.4 from [10] says that for z ∈ D , n ≥ φ ∗ n ( z ) = 0 , | φ n ( z ) /φ ∗ n ( z ) | < , and, consequently, ˜ γ n is well-defined and that | ˜ γ n | < φ n by setting λ n = 1, see (2.3). Thus from now on, φ n is the orthogonal rational function of degree n satisfying: (2.5) λ n = 1 − α n ¯ α n − | − α n ¯ α n − | φ ∗ n ( α n − ) | φ ∗ n ( α n − ) | κ n − | κ n − | = 1 . This normalization is from Langer-Lasarow [28]. It differs from the one madein Bultheel et al. [10], that corresponds to κ n = φ ∗ n ( α n ) >
0. However, inthe classical case, α n ≡
0, it is easily checked by induction, that it alsomatches the normalization k n > φ n lie in D . Theorem 2.1 impliesthat the roots of the orthogonal rational functions φ n are, in fact, in D .Another useful fact is that the OR-functions ( φ k ) ≤ k ≤ n , are orthonormal in L (cid:16) P ( .,α n ) | φ n | dm (cid:17) , see [10, Theorem 6.1.9].We already saw a definition of ORFs of the second kind (see the discussionfollowing Definition 0.1). Presently, the OR-functions of the second kind willbe introduced by an explicit formula:(2.6) ψ = 1 ,ψ n ( z ) = Z T t + zt − z ( φ n ( t ) − φ n ( z )) dµ ( t ) . Both definitions turn out to be equivalent (see Theorem 2.6 or [10, Theorem6.2.5]), but the one above is better suited for computations. The next resultwraps Lemmas 4.2.2 and 4.2.3 from [10], whose proof is a direct computationusing the orthogonality of ( φ n ). Lemma 2.3.
Let n ≥ and the function g be so that g ∗ ∈ L n − . Then ψ n ( z ) g ( z ) = Z T t + zt − z ( φ n ( t ) g ( t ) − φ n ( z ) g ( z )) dµ ( t ) . Similarly, for h such that h ∗ ∈ ζ n L n − , we have (2.7) − ψ ∗ n ( z ) h ( z ) = Z T t + zt − z ( φ ∗ n ( t ) h ( t ) − φ ∗ n ( z ) h ( z )) dµ ( t ) . Recall the Herglotz transform F µ of a measure µ defined in (0.4). Plugging h = ( B n ) ∗ in (2.7) and using that φ n is µ -orthogonal to constants, we obtainat once Proposition 2.4 ([11], Theorem 3.4) . Let φ n be the ORF of the first kindand ψ n be as in (2.6) . Then (2.8) F µ ( z ) = ψ ∗ n ( z ) φ ∗ n ( z ) + z B n ( z ) u n ( z ) φ ∗ n ( z ) , ULTIPOINT SCHUR APPROXIMATION 15 where u n is a analytic function in D given by (2.9) u n ( z ) = 2 Z T ( φ n ) ∗ ( t ) dµ ( t ) t − z , z ∈ D . In particular, ψ ∗ n /φ ∗ n interpolates F µ at 0 and at the α k for 1 ≤ k ≤ n .The theorem to come is [10, Theorem 4.2.4], with a different normaliza-tion. Theorem 2.5.
The ORFs ( φ n ) and the ( ψ n ) from (2.6) together satisfy: (2.10) (cid:20) φ n ψ n φ ∗ n − ψ ∗ n (cid:21) = p − | α n | − ¯ α n z n Y k = n (cid:20) − ˜ γ k − ˜ γ k (cid:21) (cid:20) ζ k − ( z ) 00 1 (cid:21)! (cid:20) − (cid:21) , where Π n = Q k =1 k = n p − | ˜ γ k | . In particular, ψ n is in L n . By taking determinants in (2.10), we get for z ∈ D φ n ( z ) ψ ∗ n ( z ) + φ ∗ n ( z ) ψ n ( z ) = 2 1 − | α n | (1 − α n z )( z − α n ) z B n ( z ) , and, consequently, for z ∈ T ,(2.11) φ n ( z ) ψ ∗ n ( z ) + φ ∗ n ( z ) ψ n ( z ) = 2 B n ( z ) P ( z, α n ) . Geronimus theorem.
The Geronimus-type theorem below is centralfor the whole construction. It seems first stated in [28], but it is implicitlycontained in [10, Sect. 6.4].
Theorem 2.6.
Let f ∈ S and µ be the measure associated to f by (0.4) .Then, for k ≥ , ˜ γ k +1 = γ k , where ( ˜ γ k ) are the Geronimus parameters defined in (2.2) and ( γ k ) the Schurparameters defined in (0.2) . Thus the Geronimus parameters and the Schur parameters of of a measure µ coincide. It follows from (2.10) that ( ψ n ) meets the same recurrencerelations as ( φ n ) only with Geronimus parameters − ˜ γ n rather than ˜ γ n . Thus,we see that the definition of the ORFs of the second kind given in (2.6)coincides with the one made in the introduction. Proof.
The idea is to compare the recurrence formulas (1.8) and (2.10). Weassume the sequence ( α k ) is simple, i.e. α k = α j for k = j . The proof inthe general case follows by a limiting argument. By (2.10), we have (cid:20) φ n +1 ( z ) ψ n +1 ( z ) φ ∗ n +1 ( z ) − ψ ∗ n +1 ( z ) (cid:21) = ∆ n +1 k =1 Y k = n +1 (cid:20) − (cid:21) (cid:20) γ k ˜ γ k (cid:21) (cid:20) ζ k − ( z ) 00 1 (cid:21) (cid:20) − (cid:21)! (cid:20) − (cid:21) where ∆ n +1 = p − | α n +1 | − ¯ α n +1 z Q n +1 k =1 p − | ˜ γ k | . Let now U n /V n be the n -th convergent of the Schur function with Schurparameters γ k := ˜ γ k +1 , k ≥
0. Proposition 1.6 provides us with the followingexpression for φ n , ψ n : (cid:20) φ n +1 ( z ) ψ n +1 ( z ) φ ∗ n +1 ( z ) − ψ ∗ n +1 ( z ) (cid:21) = ∆ n +1 (cid:20) − (cid:21) (cid:20) V ∗ n U ∗ n U n V n (cid:21) (cid:20) ζ
00 1 (cid:21) (cid:20) − (cid:21) (cid:20) − (cid:21) = ∆ n +1 (cid:20) zV ∗ n − U ∗ n zV ∗ n + U ∗ n − zU n + V n − zU n − V n (cid:21) . Therefore, (cid:20) φ n +1 ( z ) ψ n +1 ( z ) φ ∗ n +1 ( z ) − ψ ∗ n +1 ( z ) (cid:21) = p − | α n +1 | − ¯ α n +1 z Q n +1 k =1 p − | ˜ γ k | (cid:20) zV ∗ n − U ∗ n zV ∗ n + U ∗ n − zU n + V n − zU n − V n (cid:21) , (2.12)and(2.13) ψ ∗ n +1 φ ∗ n +1 = 1 + z U n V n − z U n V n . Consequently, U n ( z ) V n ( z ) = Ω z (cid:18) ψ ∗ n +1 ( z ) φ ∗ n +1 ( z ) (cid:19) , where Ω z ( w ) = ( w − / ( z ( w + 1)). From Proposition 2.4, we get F ( α j +1 ) = (cid:18) ψ ∗ n +1 φ ∗ n +1 (cid:19) ( α j +1 ) . Recalling that f ( z ) = Ω z ( F ( z )), it follows by Proposition 1.2 that the n + 1first Schur parameters of the function U n /V n and of the function f coincide. (cid:3) The theorem shows that the functions U n and V n are equal to the WRFs A n and B n corresponding to f . In particular, (2.12) and (2.13) imply (cid:20) φ n +1 ( z ) ψ n +1 ( z ) φ ∗ n +1 ( z ) − ψ ∗ n +1 ( z ) (cid:21) = p − | α n +1 | − ¯ α n +1 z Q n +1 k =1 p − | ˜ γ k | (cid:20) zB ∗ n − A ∗ n zB ∗ n + A ∗ n − zA n + B n − zA n − B n (cid:21) (2.14)and(2.15) ψ ∗ n +1 φ ∗ n +1 = 1 + z A n B n − z A n B n . ULTIPOINT SCHUR APPROXIMATION 17
Consequences of Geronimus theorem.
Arguing as in [22, Corol-lary 5.2], we readily see that the Schur function A n /B n corresponds to themeasure P ( .,α n +1 ) | φ n +1 | dm .The next theorem provides one with a helpful relation between the density µ ′ of the absolutely continuous part of µ , the Schur remainders ( f n ), andthe ORFs ( φ n ). It is a counterpart to Theorem 2 from [22], see also [37].Since it is heavily used in the sequel, we give the proof. Theorem 2.7.
Let ( φ n ) and ( f n ) be the ORFs and Schur remainders asso-ciated to µ and f , respectively. Then it holds a.e. on T that µ ′ = 1 − | f n | | − ζ n φ n φ ∗ n f n | P ( ., α n ) | φ n | . Proof.
From Theorem 1.9, we have on T − | f | = 1 − (cid:12)(cid:12)(cid:12)(cid:12) A n + ζ n +1 B ∗ n f n +1 B n + ζ n +1 A ∗ n f n +1 (cid:12)(cid:12)(cid:12)(cid:12) = | B n + ζ n +1 A ∗ n f n +1 | − | A n + ζ n +1 B ∗ n f n +1 | | B n + ζ n +1 A ∗ n f n +1 | . (2.16)Notice that A ∗ n B n = A n B ∗ n on T , so that ζ n +1 A ∗ n f n +1 B n + B n ζ n +1 A ∗ n f n +1 − A n ζ n +1 B ∗ n f n +1 − A n ζ n +1 B ∗ n f n +1 = 0 . Therefore, on expanding (2.16) and recalling Corollary 1.7, we find that1 − | f | = ( | B n | − | A n | )(1 − | f n +1 | ) | B n + ζ n +1 A ∗ n f n +1 | = ω n (1 − | f n +1 | ) | B n + ζ n +1 A ∗ n f n +1 | , where ω n = Q nk =0 (1 − | γ k | ).Again by Theorem 1.9, we obtain | − zf | = (cid:12)(cid:12)(cid:12)(cid:12) − zA n + ζ n +1 zB ∗ n f n +1 B n + ζ n +1 A ∗ n f n +1 (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) B n − zA n + ζ n +1 f n +1 ( A ∗ n − zB ∗ n ) B n + ζ n +1 A ∗ n f n +1 (cid:12)(cid:12)(cid:12)(cid:12) . On the other hand, Theorem 2.6 and (2.14) show zB ∗ n − A ∗ n = − ¯ α n +1 z √ −| α n +1 | √ ω n φ n +1 B n − zA n = − ¯ α n +1 z √ −| α n +1 | √ ω n φ ∗ n +1 and therefore | − zf | = ω n | − ¯ α n +1 z | − | α n +1 | (cid:12)(cid:12)(cid:12)(cid:12) φ ∗ n +1 − ζ n +1 f n +1 φ n +1 B n + ζ n +1 A ∗ n f n +1 (cid:12)(cid:12)(cid:12)(cid:12) . Recall that µ ′ ( ξ ) = (1 − | f ( ξ ) | ) / ( | − ξf ( ξ ) | ) a.e. on T . Combining allthis, we obtain µ ′ = 1 − | f n +1 | | φ n +1 | | − ζ n +1 φ n +1 φ ∗ n +1 f n +1 | − | α n +1 | | ξ − α n +1 | which achieves the proof. (cid:3) Weighted L -convergence of Schur remainders The material reviewed so far is known, and it is meant as a preparationfor the forthcoming results which are new. As we start doing analysis ratherthan algebra, the assumptions (0.3) and (0.11)-(0.13) or (0.14), (0.15), willstart playing a key role.We begin quoting a lemma which is [10, Theorem 9.7.1].
Lemma 3.1.
Assuming (0.3) , we get in the weak-* convergence of measures ( ∗ ) − lim n P ( ., α n ) | φ n | dm = dµ. The two theorems below address the L -convergence of Schur remaindersunder different assumptions. Recall Acc ( α k ) is the set of accumulationpoints of ( α k ). Theorem 3.2.
Let (0.3) be in force and lim k | α k | = 1 . Assume that (0.11) - (0.13) hold. Then (3.1) lim k Z T | f k | P ( ., α k ) dm = 0 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then one can replace (0.11) and (0.12) with (0.14) .Proof. It is enough to prove (3.1) for any subsequence ( α n k ), converging to α ∈ Acc ( α k ). For simplicity, the subsequence is still denoted by ( α k ).By Theorem 2.7, we get | φ n | µ ′ (1 + | f n | − Re ( ζ n φ n φ ∗ n f n )) = (1 − | f n | ) P ( ., α n )and, consequently, | f n | = P ( ., α n ) − | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ + 2 | φ n | µ ′ Re ( ζ n φ n φ ∗ n f n ) P ( ., α n ) + | φ n | µ ′ . Hence, we obtain | f n | = P ( ., α n ) − | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ − P ( ., α n ) − | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ Re (cid:18) ζ n φ n φ ∗ n f n (cid:19) + Re (cid:18) ζ n φ n φ ∗ n f n (cid:19) . Since ζ n ( α n ) = 0, we get by harmonicity Z T Re (cid:18) ζ n φ n φ ∗ n f n (cid:19) P ( ., α n ) dm = 0 , and Z T | f n | P ( ., α n ) dm = Z T P ( ., α n ) − | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ (cid:18) − Re (cid:18) ζ n φ n φ ∗ n f n (cid:19)(cid:19) P ( ., α n ) dm. Obviously, (cid:12)(cid:12)(cid:12)(cid:12) − Re (cid:18) ζ n φ n φ ∗ n f n (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T | f n | P ( ., α n ) dm ≤ Z T (cid:12)(cid:12)(cid:12)(cid:12) − | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ (cid:12)(cid:12)(cid:12)(cid:12) P ( ., α n ) dm. ULTIPOINT SCHUR APPROXIMATION 19
Let(3.3) g n = 2 | φ n | µ ′ P ( ., α n ) + | φ n | µ ′ . Using that 4 x / (1 + x ) ≤ x for x ≥
0, we deduce Z T g n P ( ., α n ) dm = Z T | φ n | µ ′ P ( ., α n ) − ) (1 + | φ n | µ ′ P ( ., α n ) − ) P ( ., α n ) dm ≤ Z T | φ n | µ ′ P ( ., α n ) − P ( ., α n ) dm = Z T | φ n | µ ′ dm ≤ Z T | φ n | dµ = 1 . Therefore, by the Schwarz inequality, it follows that(3.4) Z T g n P ( ., α n ) dm ≤ (cid:18)Z T g n P ( ., α n ) dm (cid:19) / ≤ . Furthermore, again by the Schwarz inequality, Z T p µ ′ P ( ., α n ) dm = Z T √ | φ n |√ µ ′ p P ( ., α n ) p P ( ., α n ) + | φ n | µ ′ p P ( ., α n ) + | φ n | µ ′ p P ( ., α n ) √ | φ n | dm ≤ (cid:18)Z T g n P ( ., α n ) dm (cid:19) / (cid:18) Z T (cid:18) P ( ., α n ) | φ n | + µ ′ (cid:19) P ( ., α n ) dm (cid:19) / . Recall that the ORFs ( φ k ) ≤ k ≤ n are orthonormal in L (cid:16) P ( .,α n ) | φ n | dm (cid:17) and,consequently, Z T f P ( ., α n ) | φ n | dm = Z T f dµ for f ∈ L n + L n . Obviously, P ( z, α n ) = z/ ( z − α n ) + ¯ α n z/ (1 − ¯ α n z ) lies inthe latter space and(3.5) Z T P ( ., α n ) P ( ., α n ) | φ n | dm = Z T P ( ., α n ) dµ. Using (3.5), we arrive at(3.6) Z T p µ ′ P ( ., α n ) dm ≤ (cid:18)Z T g n P ( ., α n ) dm (cid:19) / (cid:18)Z T P ( ., α n ) dµ (cid:19) / . Recall now that ( α n ) converges to α ∈ T . By hypothesis, µ ′ is continuousat α and there is no singular component µ s in a neighborhood of this point.Thus, passing to the inferior limit in (3.6), we obtain p µ ′ ( α ) ≤ p µ ′ ( α ) lim inf n (cid:18)Z T g n P ( ., α n ) dm (cid:19) / . Moreover, by Fatou’s theorem, the same conclusion holds if we assume (0.14)instead of (0.11)-(0.12) provided that ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T . Therefore, since µ ′ ( α ) > n Z T g n P ( ., α n ) dm ≥ . Combining this inequality with (3.4), we see that(3.7) lim n Z T g n P ( ., α n ) dm = lim n Z T g n P ( ., α n ) dm = 1 , and subsequently thatlim n Z T (1 − g n ) P ( ., α n ) dm = Z T P ( ., α n ) dm − n Z T g n P ( ., α n ) dm + lim n Z T g n P ( ., α n ) dm = 0 . With the Schwarz inequality and (3.2), we finish the proof of the first partof the theorem. (cid:3)
Remark 3.3.
As a partial converse, (3.1) implies that | f | < a.e. on Acc ( α k ) ∩ T . Indeed, observe if | f | = 1 a.e. on E ⊂ Acc ( α k ) ∩ T , | E | >
0, that | f n | = 1a.e. on E by Theorem 2.7. The Lebesgue’s theorem says that the set ofdensity points of E coincides with E up to a set of Lebesgue measure zero.Comparing the Poisson kernel and the box kernel, we see that the integralin (3.1) cannot go to zero if we pick for α a density point of E .A similar convergence holds when the ( α n ) are compactly included in D .The statement below may seem strange, since the Poisson kernel P ( ., α n )is bounded from above and below and therefore superfluous. However, itis convenient to prove the theorem in this form to have it team up withTheorem 3.2 in order to produce Corollary 3.5. Theorem 3.4.
Let the sequence ( α k ) be compactly included in D . Then, | f | < a.e. on T if and only if (3.8) lim n Z T | f n | P ( ., α n ) dm = 0 . Proof.
The “if” part is trivial since | f n | = 1 wherever | f | = 1, so we focuson the “only if”. As a preliminary, notice that if I is an open arc on T suchthat µ has no mass at the end-points of I , it holds that(3.9) lim sup n Z I P ( ., α n ) | φ n | dm ≤ µ ( I ) . Indeed, in this case, any nested sequence of open arcs I m decreasing to I is such that lim m µ ( I m ) = µ ( I ) = µ ( I ). Therefore by the Tietze-Urysohntheorem, there is to each ε > h I ∈ C ( T ) such that h I = 1 on I and R T h I dµ ≤ µ ( I ) + ε . Obviously Z I P ( ., α n ) | φ n | dm ≤ Z T h I P ( ., α n ) | φ n | dm, and using Lemma 3.1lim n Z T h I P ( ., α n ) | φ n | dm = Z T h I dµ ≤ µ ( I ) + ε. Since ε was arbitrary, this settles the preliminary. Next, define g n as in(3.3). Arguing as in the previous theorem, we see that equation (3.4) still ULTIPOINT SCHUR APPROXIMATION 21 holds. Now, it is enough to show that the conclusion of the theorem holdsfor some infinite subsequence of each sequence of integers. Thus, by Helly’stheorem, we are left to establish (3.8) along a subsequence n k such that α n k → α ∈ Acc ( α k ), α ∈ D , and having the property that g n k convergesto g ∈ L ∞ ( T ) in the ∗ -weak sense. Clearly 0 ≤ g ≤ g n k . Pick ξ ∈ T a Lebesgue point of both g and µ , and let ( I m ) be anested sequence of open arcs decreasing to { ξ } such that µ has no mass atthe end-points of any I m . For each m , by the Schwarz inequality,1 | I m | Z I m p µ ′ dm = 1 | I m | Z I m √ | φ n k |√ µ ′ p P ( ., α n k ) + | φ n k | µ ′ p P ( ., α n k ) + | φ n k | µ ′ √ | φ n k | dm ≤ (cid:18) | I m | Z I m g n k dm (cid:19) / (cid:18) | I m | Z I m (cid:18) P ( ., α n k ) | φ n k | + µ ′ (cid:19) dm (cid:19) / . (3.10)Passing to the limit in (3.10) as n k → ∞ and using (3.9), we obtain1 | I m | Z I m p µ ′ dm ≤ (cid:18) | I m | Z I m gdm (cid:19) / (cid:18) µ ( I m ) | I m | + 12 | I m | Z I m µ ′ dm (cid:19) / . Letting now m → ∞ yields(3.11) p µ ′ ( ξ ) ≤ p g ( ξ ) (cid:18) µ ′ ( ξ ) + 12 µ ′ ( ξ ) (cid:19) / ≤ p g ( ξ ) p µ ′ ( ξ ) . By Lebesgue’s theorem almost every ξ ∈ T satisfies our requirements, andfrom our assumption that | f | < µ ′ >
0, a.e. on T . Conse-quently g ≥ g = 1, a.e. on T . Recalling thatlim n P ( ., α n ) = P ( ., α ) uniformly on T , we obtain (3.7) from (3.4) and con-clude as in Theorem 3.2. (cid:3) Corollary 3.5.
Let (0.3) , (0.11) - (0.13) hold and | f | < a.e. on T . Then lim k Z T | f k | P ( ., α k ) dm = 0 . When ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , assumptions (0.11) - (0.12) may be replaced by (0.14) .Proof. It is readily checked that Theorems 3.2 and 3.4 remain valid forsubsequences. If the corollary did not hold, it would contradict one of them. (cid:3)
A closer look at the proof of Theorem 3.2 shows that assumption (0.13)is not really necessary. If α ∈ Acc ( α k ) ∩ T and lim k α k = α , all we need islim k Z T P ( ., α k ) dµ s = 0 . For instance if µ s is a Dirac mass at α and the α k converge tangentially to α , this could still hold. Convergence of Wall rational functions A n /B n We now discuss different kinds of convergence for the WRFs. This isessentially an interpretation of the results in the previous section, exceptthat we appeal at some point to Proposition 5.6 and Theorem 5.8. Thereader will easily convince himself that there is no loophole, i.e. that thesedo not use any result of the present section.4.1.
Convergence on compact subsets and w.r.t. pseudohyperbolicdistance.
Let us begin with an old result that goes back to [46].
Theorem 4.1.
Let (0.3) hold. Then A n /B n converges to f uniformly oncompact subsets of D .Proof. As ( A n /B n ) is a family of Schur functions, it is normal. Thereforea subsequence that converges uniformly on compact subsets of D can beextracted from any subsequence. Let g be the limit of such a subsequence.As ( A n /B n )( α k ) = f ( α k ) for all ≤ n + 1, f ( α k ) = g ( α k ) for all k . So, thefunction f − g ∈ H ∞ vanishes on ( α k ) hence it is zero by assumption (0.3).Thus, f is the only limit point. (cid:3) Recall that the pseudohyperbolic distance ρ on D is defined by ρ ( z, w ) = | z − w | / | − ¯ wz | and it is trivially invariant under M¨obius transforms of D . Theorem 4.2.
Under the assumptions of Corollary 3.5, it holds that lim n Z T ρ (cid:18) f, A n B n (cid:19) P ( ., α n +1 ) dm = 0 . Proof.
The invariance of the pseudohyperbolic distance under M¨obius trans-forms and relations (1.3) and (1.4) show that ρ (cid:18) f, A n B n (cid:19) = ρ ( τ ◦ · · · ◦ τ n ( f n +1 ) , τ ◦ · · · ◦ τ n (0)) = ρ ( f n +1 ,
0) = | f n +1 | . Corollary 3.5 finishes the proof. (cid:3)
Convergence w.r.t. the hyperbolic metric.
In the disk, the hy-perbolic metric is defined by(4.1) P ( z, ω ) = log (cid:18) ρ ( z, ω )1 − ρ ( z, ω ) (cid:19) . Here is an analogue of the “only if” part of Theorem 2.6 from [22].
Theorem 4.3.
Let (0.3) , (0.11) - (0.13) be in force, and µ ∈ (S) . Then lim n Z T P (cid:18) f, A n B n (cid:19) P ( ., α n +1 ) dm = 0 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then it is enough toassume instead of (0.11) and (0.12) that (0.15) holds.Proof. We already saw that ρ ( f, A n /B n ) = | f n +1 | whence(4.2) P (cid:18) f, A n B n (cid:19) = log (cid:18) | f n +1 | − | f n +1 | (cid:19) . ULTIPOINT SCHUR APPROXIMATION 23
By Theorem 2.7,(4.3) | φ ∗ n | | S | | − ¯ α n ξ | − | α n | = 1 − | f n | | − ζ n φ n φ ∗ n f n | , a.e. on T . If g is a Schur function, then 1 − g ∈ H ∞ and Re (1 − g ) > − g is an outer function in H ∞ ( D ) (see [18], Corollary 4.8).Consequently, Z T log | − g | P ( ., α n ) dm = log | − g ( α n ) | , and, putting g = ζ n φ n φ ∗ n f n , we get Z T log | − ζ n φ n φ ∗ n f n | P ( ., α n ) dm = 0 . Using the previous equality and (4.3), we see that Z T log (cid:18) | φ ∗ n | | S | | − ¯ α n ξ | − | α n | (cid:19) P ( ξ, α n ) dm ( ξ ) = Z T log(1 −| f n | ) P ( ξ, α n ) dm ( ξ ) . Since log | φ ∗ n | , log | S | , and log | − ¯ α n ξ | are harmonic in D , we continue as(4.4) log (cid:0) | φ ∗ n ( α n ) | | S ( α n ) | (1 − | α n | ) (cid:1) = Z T log(1 − | f n | ) P ( ., α n ) dm, and, by Theorem 5.8 to come, we deduce that(4.5) lim n Z T log(1 − | f n | ) P ( ., α n ) dm = 0 . Since log(1 + x ) ≤ x for x > −
1, we have0 ≤ | f n | ≤ − log(1 − | f n | ) , ≤ log(1 + | f n | ) ≤ | f n | . Therefore, by the first inequality above and (4.5),lim n Z T | f n | P ( ., α n ) dm = 0 . From this, with the help of the second inequality and the Schwarz inequality,lim n Z T log(1 + | f n | ) P ( ., α n ) dm = 0 . Since log(1 − | f n | ) = log(1 − | f n | ) + log(1 + | f n | ), we now see thatlim n Z T log(1 − | f n | ) P ( ., α n ) dm = 0 . Referring to (4.2), we finish the proof. (cid:3)
Convergence in L ( T ) . The next theorem follows easily from Corol-lary 3.5. We begin with
Lemma 4.4.
For z ∈ T , we have (cid:12)(cid:12)(cid:12)(cid:12) f ( z ) − A n B n ( z ) (cid:12)(cid:12)(cid:12)(cid:12) = | f n +1 ( z ) | (cid:12)(cid:12)(cid:12)(cid:12) − A n B n ( z ) f ( z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | f n +1 ( z ) | . Proof.
The equality follows from ξ n +1 f n +1 = ( A n − B n f ) / ( f A ∗ n − B ∗ n ) whichis inverse to (1.9). The inequality follows because f and A n /B n are Schur. (cid:3) Theorem 4.5.
The limiting relation (4.6) lim n Z T (cid:12)(cid:12)(cid:12)(cid:12) f − A n B n (cid:12)(cid:12)(cid:12)(cid:12) p P ( ., α n +1 ) dm = 0 holds in the following cases: (1) if p ≥ under the assumptions of Corollary 3.5, (2) for ≤ p < ∞ if Acc ( α k ) ∩ T = ∅ and | f | < a.e. on T .Proof. This is immediate from Lemma 4.4, Corollary 3.5, the fact that | f n | ≤
1, the existence of pointwise a.e. converging subsequences in L -convergentsequences, and the dominated convergence theorem. (cid:3) Uniform convergence.
When µ is sufficiently smooth, the previous L p -convergence is uniform. Theorem 4.6.
Let (0.3) hold and dµ = µ ′ dm be absolutely continuous on T .Assume that µ ′ ∈ W − /p,p ( T ) with p > . If µ ′ > on some neighborhood O ( Acc ( α k ) ∩ T ) , then (4.7) lim n (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) f − A n B n (cid:19) p P ( ., α n +1 ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ = 0 . Proof.
From (0.4) and (2.15), one easily computes f ( z ) − A n /B n ( z ) = 2 F µ φ ∗ n +1 ( z ) − ψ ∗ n +1 ( z ) z (1 + F µ ( z ))( ψ ∗ n +1 ( z ) + φ ∗ n +1 ( z )) . Besides, we observe from (2.11) that (cid:12)(cid:12) φ ∗ n +1 ( z ) + ψ ∗ n +1 ( z ) (cid:12)(cid:12) = (cid:12)(cid:12) φ ∗ n +1 ( z ) (cid:12)(cid:12) + (cid:12)(cid:12) ψ ∗ n +1 ( z ) (cid:12)(cid:12) + 2 P ( z, α n +1 ) , z ∈ T . From this and the fact that Re F µ ≥
0, we obtain on T that | ( f − A n /B n ) p P ( ., α n +1 ) | ≤ √ | F µ φ ∗ n +1 − ψ ∗ n +1 | . Observing that µ ′ is continuous on T since p >
4, we may apply Corollary5.14 to the effect that F µ φ ∗ n − ψ ∗ n converges to zero uniformly on T . (cid:3) A Szeg˝o-type problem
In this final section, we study the asymptotic behavior of ORFs lookingfor an analogue of the Szeg˝o theorem in the rational setting when ( α k )may approach T . The results are of a novel type and, we hope, worthy forthemselves. We need them also to complete the proofs of Theorems 4.3 and4.6.5.1. Preliminaries.
With π n defined as in (0.9), we denote by P n (cid:0) dµ/ | π n | (cid:1) ⊂ L (cid:0) dµ/ | π n | (cid:1) the subspace of polynomials of degree at most n . The space H (cid:0) dµ/ | π n | (cid:1) is the closure of all polynomials in L (cid:0) dµ/ | π n | (cid:1) . The repro-ducing kernels of H (cid:0) dµ/ | π n | (cid:1) and P n (cid:0) dµ/ | π n | (cid:1) , are denoted by E n and R n , respectively. Since there will be several measures involved, we indicatethe dependence in square brackets when necessary. For example, we maywrite φ n [ µ ] , E n [ µ ] , R n [ µ ], or S [ µ ], see (0.16). We also put dµ n := dµ/ | π n | . ULTIPOINT SCHUR APPROXIMATION 25
Proposition 5.1.
Let µ ∈ (S) be absolutely continuous. Then, E n [ µ ]( ξ, ω ) = 11 − ξ ¯ ω π n ( ξ ) π n ( ω ) S ( ξ ) S ( ω ) . The proof is straightforward and stems from the density of polynomialsin H (cid:0) dµ/ | π n | (cid:1) , the Cauchy formula, and the identity | S | = µ ′ on T . Proposition 5.2.
The following identity holds: (5.1) | π n φ ∗ n | = | R n ( ., α n ) |k R n ( ., α n ) k L ( dµ n ) . Proof.
Let p n − be a polynomial of degree at most n −
1. As φ n is orthogonalto L n − , we have Z T φ n p n − π n − dµ = 0 . On the other hand, since φ n = ( φ n ) ∗ and 1 /t = ¯ t on T , Z T φ n p n − π n − dµ = Z T φ ∗ n ( t ) π n ( t ) t n π n ( t ) p n − ( t )(1 − ¯ α n t ) π n ( t ) dµ ( t )= Z T π n ( t ) φ ∗ n ( t )¯ t n − p n − ( t )(¯ t − ¯ α n ) dµ ( t ) | π n ( t ) | = Z T π n ( t ) φ ∗ n ( t ) t n − p n − (cid:18) t (cid:19) ( t − α n ) ! dµ ( t ) | π n ( t ) | . As t n − p n − (1 / ¯ t ) ranges over P n − ( z ) when p n − does, π n φ ∗ n is µ n -orthogo-nal to every polynomial of degree ≤ n that vanishes at α n . This is also true of R n ( ., α n ), hence π n φ ∗ n and R n ( ., α n ) are proportional. Since the right-handside of (5.1) and π n φ ∗ n have unit norm in L ( dµ n ), we are done. (cid:3) In the following corollary µ is not necessarily absolutely continuous. Corollary 5.3.
For µ ∈ (S) and n ≥ , we have that (5.2) | φ ∗ n ( α n ) | | S ( α n ) | (1 − | α n | ) = R n ( α n , α n ) E n [ µ ac ]( α n , α n ) ≤ . Proof.
By elementary properties of reproducing kernels, we get k R n ( ., α n ) k L ( dµ n ) = R n ( α n , α n ) , and k E n ( ., α n ) k L ( dµ n ) = E n ( α n , α n ) . Therefore, from Proposition 5.2, we obtain | π n ( α n ) φ ∗ n ( α n ) | = | R n ( α n , α n ) | k R n ( ., α n ) k L ( dµ n ) = R n ( α n , α n )hence the equality in (5.2) from the formula for E n [ µ ac ] in Proposition 5.1.Observing now that || . || L ( dµ ac ) ≤ || . || L ( dµ ) , we get a contractive injection H (cid:0) dµ/ | π n | (cid:1) ⊂ H (cid:0) dµ ac / | π n | (cid:1) , from which it follows easily that E n [ µ ]( w, w ) ≤ E n [ µ ac ]( w, w ), for w ∈ D . Since R n ( ., α n ) is the orthogonal projection of E n [ µ ]( ., α n ) on P n (cid:0) dµ/ | π n | (cid:1) k R n ( ., α n ) k L ( dµ n ) ≤ k E n [ µ ]( ., α n ) k L ( dµ n ) , and therefore R n ( α n , α n ) E n [ µ ac ]( α n , α n ) ≤ R n ( α n , α n ) E n [ µ ]( α n , α n ) ≤ , as desired. (cid:3) It is a well-known theorem of Beurling that [18, Ch. II, Theorem 7.1] thatfunctions of the form Sp , with p a polynomial, are dense in H ( D ) when S is outer. We shall need a local refinement of this result (compare to [18, Ch.2, Theorem 7.4]) where, in addition, polynomials get replaced by functionsin L n . Lemma 5.4.
Let (0.3) hold and O be open in T . Let S ∈ H ( D ) be anouter function which is continuous on O with | S | > δ > there. Then, toevery compact K ⊂ O , there is a sequence of rational functions R m ∈ L m such that(i) k − R m S k → as m → ∞ ,(ii) the functions − R m S go to zero uniformly on K .Proof. Recall that log | S | ∈ L ( T ) and put u n = min { a n , − log | S |} , where a n > ∞ so fast that(5.3) ∞ X n =0 (cid:18) − exp (cid:18)Z T ( u n + log | S | ) dm (cid:19)(cid:19) < ∞ . Let S n be the outer function such that | S n | = e u n on T , normalized so that S n (0) >
0. Then S n ∈ H ∞ ( D ) and | S n S | ≤ T with | S n S | = 1 on O for n large enough, therefore we can write(5.4) SS n ( z ) = exp Z T \O t + zt − z log | SS n | dm ( t ) ! , showing that SS n extends analytically across O to an analytic function G n on C \ ( T \ O ). Moreover, ( G n ) is a normal family since | log | S n || ≤ | log | S || on T . Besides, expanding k − S n S k and using (5.3), we obtain(5.5) ∞ X n =0 k − S n S k ≤ ∞ X n =0 (1 − S n (0) S (0)) < ∞ so that, by the Borel-Cantelli lemma, SS n converges to 1 a.e. on T . Then,by normality, SS n converges to 1 locally uniformly on O .Next, fix a compact K ⊂ O and let S n,r ( z ) := S n ( rz ) for 0 < r < S n,r = P rz ∗ S n , and since S n ∈ L ∞ ( T ) is continuous on O where itequals G n /S , it follows from standard properties of Poisson integrals [18,Ch. 2] that S n,r converges to S n boundedly pointwise a.e. on T and locallyuniformly on O as r →
1. In particular, S n,r S converges to S n S in L ( T )for fixed n as r →
1. Hence to each n there is r n such that, say, (cid:26) k − S n,r n S k < k − S n S k + 2 − n , sup K | S n,r n − S n | < /n. ULTIPOINT SCHUR APPROXIMATION 27
Clearly S n,r n lies in A ( D ), therefore is can be uniformly approximated on T by functions from ∪ k L k since (0.3) holds. Therefore, to each n , there is aninteger m n and R m n ∈ L m n such that(5.6) (cid:26) k − R m n S k < k − S n S k + 2 − n , sup K | R m n − S n | < /n. Without loss of generality, we assume that m n strictly increases with n .Now, by (5.5) and since | SS n | ∈ L ∞ ( T ), the first relation in (5.6) implies ∞ X n =0 k − R m n S k < ∞ whence R m n S converges to 1 in H ( D ) as n → ∞ . In another connection, | − R m n S | ≤ | − S n S | + | R m n − S n || S | and the second relation in (5.6) yields that R m n S converges uniformly to 1on K when m n → ∞ . To complete the proof, it remains to put R m = R m k where k is the greatest integer such that m k ≤ m . (cid:3) An a priori bound on ORFs.
We derive in this subsection a priori estimates for ORFs akin to the classical bounds for orthogonal polynomials[20, Ch. 4, Theorems 4.6, 4.8], [44, Ch. 12, Theorem 12.1.3]. These in factare new even in the classical polynomial case, as they yield information incases where µ ′ vanishes, thereby generalizing some of the results from [40].Their proof rely on basic properties of the Sobolev spaces W ,p (Ω). For1 < p < ∞ and Ω ⊂ C an open set with boundary ∂ Ω, recall that W ,p (Ω) = { f ∈ L p (Ω) : || f || L p (Ω) + || f ′ || L p (Ω) < ∞} , where the derivatives are understood in the distributional sense. If ∂ Ω ispiecewise smooth and D indicates the space of C ∞ functions with compactsupport in C , then the restriction D| Ω is dense in W ,p (Ω). For g ∈ W ,p (Ω)and ( g n ) a sequence in D| Ω converging to g , one can show that the trace of g n on ∂ Ω converges in W − /p,p (Ω), see (0.19). This allows one to definethe trace of g ∈ W ,p (Ω) on ∂ Ω as a member of W − /p,p ( ∂ Ω). With thisdefinition, Stokes’ formula holds for Sobolev differential forms just like itdoes for smooth ones.We put η = x + iy and use the standard notation ∂∂η = 12 (cid:18) ∂∂x − i ∂∂y (cid:19) , ∂∂ ¯ η = 12 (cid:18) ∂∂x + i ∂∂y (cid:19) . The usual rules of differentiation apply to ∂/∂η , ∂/∂ ¯ η , and the relation ∂V /∂ ¯ η = 0 means that V is analytic. We need a function-theoretic lemma. Lemma 5.5.
Let I ⊂ T be an open arc and Ω ⊂ D an open set such that Ω ∩ T ⊂ I . If g ∈ H ( D ) is such that g | I ∈ W − /p,p ( I ) for some < p < ∞ ,then g | Ω ∈ W ,p (Ω) .Proof. The restriction g | I extends to a function h ∈ W − /p,p ( T ) [1, 7.69].By standard elliptic regularity, there is a harmonic function U ∈ W ,p ( D )such that U | T = h , where the trace is understood in the Sobolev sense [12].Any harmonic conjugate V of U in turn belongs to W ,p ( D ) for ∂V /∂ ¯ η = i∂U/∂ ¯ η by the Cauchy-Riemann equations. Hence the analytic function F = U + iV lies in W ,p ( D ), and it follows from the definition that F ( rη ) → F ( η ) in W ,p ( D ) as r → − . Thus by the trace theorem, the restrictionof F to every circle T r centered at 0 of radius r < W − /p,p ( T r )norm at most C k F k W ,p ( D ) for some constant C [1, 7.39]. A fortiori then, F ∈ H p ( D ) and its Sobolev trace on T must coincide with its nontangentiallimit. Consequently g − F is a H ( D )-function which is pure imaginary on I , therefore it extends analytically by reflection across I . In particular g − F is smooth on a neighborhood of Ω and g = F + ( g − F ) lies in W ,p (Ω). (cid:3) Our a priori bound will depend on the connection between φ n , ψ n and u n defined in (2.9). By Proposition 2.4, zu n ( z ) = F µ ( z )( φ n ( z )) ∗ − ( ψ n ( z )) ∗ for z ∈ T . Multiplying by φ n and taking real parts, we get for z ∈ T µ ′ ( z ) | φ n ( z ) | = Re(( ψ n ) ∗ ( z ) φ n ( z )) + Re( zφ n ( z ) u n ( z )) , and a short computation using (2.11) gives us (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( z ) φ n ( z ) − z ¯ u n ( z )2 (cid:12)(cid:12)(cid:12)(cid:12) = µ ′ ( z ) P ( z, α n ) + (cid:12)(cid:12)(cid:12)(cid:12) u n ( z )2 (cid:12)(cid:12)(cid:12)(cid:12) . Thus, either | µ ′ ( z ) φ n ( z ) | ≤ | u n ( z ) | or | µ ′ ( z ) φ n ( z ) | / < µ ′ ( z ) P ( z, α n ) + | u n ( z ) | /
4. Therefore, for z ∈ T ,(5.7) µ ′ ( z ) | φ n ( z ) | ≤ | u n ( z ) | + µ ′ ( z ) P ( z, α n ) . Proposition 5.6.
Let µ ∈ (S) and I ⊂ T be an open arc disjoint from supp µ s . Assume that S ∈ W − /p,p ( I ) with / < p < ∞ . Then, to eachcompact K ⊂ I ,i) there is a neighborhood O ( K ) in I such that u n | O ( K ) ∈ W − /γ,γ ( O ) for < γ < p/ ( p + 4) , with norm depending on µ , K , and γ only.ii) If moreover p > , then | u n | ≤ C on O ( K ) and u n ∈ H s ( O ( K )) for < s < ( p − / p , where C and the H¨older constant depend on µ , K , and s only; in particular, from (5.7) , we obtain for ξ ∈ K (5.8) µ ′ ( ξ ) | φ n ( ξ ) | ≤ C + µ ′ P ( ξ, α n ) . Proof.
We may assume that K = T , otherwise the conclusion follows uponwriting T = K ∪ K . Let J = ( e iθ , e iθ ) be an open arc compactly includedin I and containing K . Fix 0 < ε < c := [ e iθ , (1 + ε ) e iθ ], c := [(1 + ε ) e iθ , e iθ ], and the circular arc c = { (1 + ε ) e iθ : θ ≤ θ ≤ θ } . Let C = c ∪ c ∪ c be the open contour joining e iθ to e iθ . Orient the Jordan curve Γ = C ∪ J counterclockwise, and letΩ denote its interior. Put Ω = { z ∈ D ; 1 / ¯ z ∈ Ω } for the reflected set.Lemma 5.5 implies that S ∈ W ,p (Ω), hence G ( z ) := S (1 / ¯ z ) and H ( z ) := S (1 / ¯ z ) belong to W ,p (Ω ). By a classical estimate [16, Theorem 5.4], H ( D )embeds continuously in L β ( D ) for 2 < β < A fortiori then, S ∈ L β (Ω)whence G, H ∈ L β (Ω ) by reflection. In particular, from the Leibnitz ruleand H¨older’s inequality, it follows since p > / GH ( z ) = | S (1 / ¯ z ) | liesin W ,α (Ω ) for some α >
1. Pick z / ∈ Ω and apply Stokes’ theorem on Γto the differential form GH ( η )( φ n ) ∗ ( η ) / ( η ( η − z )) dη :(5.9) Z C∪ J GH ( η )( φ n ) ∗ ( η ) η dηη − z = − Z Ω ( ∂G/∂ ¯ η )( η ) η − z H ( φ n ) ∗ ( η ) η dη ∧ d ¯ η, ULTIPOINT SCHUR APPROXIMATION 29 where we took into account that H ( φ n ) ∗ ( η ) / ( η ( η − z )) is analytic on Ω ,since H and ( φ n ) ∗ are analytic on C \ D while 0 , z / ∈ Ω .As GH ( ξ ) dξ/ξ = idµ ( ξ ) on ¯ J because supp µ s ∩ I = ∅ by assumption, wededuce from (5.9) and (2.9) that, for z ∈ D (5.10) u n ( z ) = 2 Z T \ J ( φ n ) ∗ ( ξ ) dµ ( ξ ) ξ − z − i Z C GH ( ξ )( φ n ) ∗ ( ξ ) ξ dξξ − z − i Z Ω ( ∂G/∂ ¯ η )( η ) η − z H ( φ n ) ∗ ( η ) η dη ∧ d ¯ η. On any O ( K ) where z remains at strictly positive distance from T \ J ,the first integral in the right-hand side of (5.10) is uniformly bounded andsmooth by the Schwarz inequality because || ( φ n ) ∗ || L ( dµ ) = 1 (recall that | ( φ n ) ∗ | = | φ n | on T ).Next, since φ n S ∈ H ( D ) and || φ n S || = || φ n || L ( dµ ac ) ≤
1, it follows fromthe Fej`er-Riesz inequality [16, Theorem 3.13] that the L -norm of φ n S overany diameter of D is at most 1 / √
2. Also, the L -norm of φ n S over thecircle centered at zero of radius 1 / (1 + ε ) is less than 1. Hence, by reflectionacross T , the L -norm of H ( φ n ) ∗ on C is uniformly bounded. Moreover,since G ∈ W ,p (Ω ), the trace theorem implies that its restriction to C liesin W − /p,p ( C ). By the embedding theorem for Besov spaces [1, Theorem7.34], this restriction belongs to L p/ (2 − p ) ( C ) if p <
2, to each L q ( C ) with 1 ≤ q < ∞ if p = 2, and it is bounded if p >
2. Thus from H¨older’s inequality, itfollows since p > / L ( C )-norm of G | C it at most C k G k W ,p (Ω ) ,where C is a constant depending only of p and C . Consequently, by theCauchy-Schwarz inequality, the second integral in the right-hand side of(5.10) is uniformly bounded and smooth on any O ( K ) remaining at positivedistance from C .We turn to the third integral, which is taken with respect to two-dimensionalLebesgue measure since dη ∧ d ¯ η = − idxdy . For z ∈ Ω , observe that thefunction V ( z ) = Z Ω v ( η ) η − z dη ∧ d ¯ η lies in W ,γ (Ω ) whenever v ∈ L γ (Ω ) with 1 < γ < ∞ . Indeed, if d is thediameter of Ω , it holds that k V k L γ (Ω ) ≤ d k v k L γ (Ω ) [6, Theorem 4.3.12]while the distributional derivatives ∂V /∂ ¯ z and ∂V /∂z equal respectively to v and the restriction to Ω of S ˇ v , where ˇ v is the extension of v by 0 tothe whole of C and S indicates the Beurling transform [6, Theorem 4.3.10].Since the latter is a bounded operator on L γ ( C ) [6, Theorem 4.5.3], it followsthat V ∈ W ,γ (Ω ) with norm depending on Ω and k v k L γ (Ω ) . Apply thisto v ( η ) = H ( η )( φ n ) ∗ ( η ) η ∂G∂ ¯ η ( η )so that V becomes the third integral in (5.10), up to the factor − i . Onthe one hand, G ∈ W ,p (Ω ) so that ∂G/∂ ¯ η ∈ L p (Ω ). On the other hand,we pointed out already that H ( D ) embeds in L β ( D ) for 2 < β <
4, hencethe L β (Ω)-norm of φ n S is uniformly bounded and so is the L β (Ω )-norm It belongs in fact to the Lorentz space L p/ (2 − p ) ,p ( C ) that we did note introduce; thelatter is included in L p/ (2 − p ) ( C ) because p < p/ (2 − p ) since p >
1, see [47, Lemma 1.8.13]. of H ( φ n ) ∗ by reflection. Therefore by H¨older’s inequality, v ∈ L γ (Ω ) with1 /γ = 1 /p + 1 /β . As p > /
3, this allows us to pick γ arbitrarily in therange (1 , p/ ( p + 4)), and assertion i ) now follows from the trace theorem.If p > < γ < p/ ( p +4), so assertion ii ) is a consequence of(5.7) and the fact that W − /γ,γ ( I ) embeds continuously in H − /γ ( I ). (cid:3) The importance of the above proposition lies with the fact that the boundsare independent of n and ( α k ), except for the presence of P ( ., α n ) in (5.8)(which reduces to 1 in the classical case).It is useful to know conditions on µ ′ = | S | for Proposition 5.6 to apply.Here is a simple criterion. Lemma 5.7.
Let < p < ∞ and µ ∈ (S) . For I ⊂ T an open arc,if µ ′ | I ∈ W − /p,p ( I ) and < δ ≤ µ ′ ( t ) ≤ M < ∞ on I , it holds that S | J ∈ W − /p,p ( J ) for each relatively compact subarc J ⊂ I .Proof. Let ϕ ∈ W − /p,p ( T ) coincide with µ ′ / I with 0 < δ ′ ≤ ϕ ≤ M ′ < ∞ ; such an extension is easily constructed by reflexion across theendpoints of I , see [21, Theorem 1.5.2.3]. As ϕ ≥ δ ′ >
0, we get that log ϕ ∈ W − /p,p ( I ) for log is Lipschitz continuous on the range of ϕ . Therefore H = log ϕ + i ˇ(log ϕ ) ∈ W − /p,p ( T ), and so does S = exp H because exp isLipschitz continuous on the range of H since ϕ ≤ M ′ . Now, S is an outerfunction having the same modulus as S on I , therefore we see as in (5.4)that S/S extends analytically across I . This entails that S ∈ W − /p,p ( J )whenever J is relatively compact in I . (cid:3) It is straightforward to check that the product of bounded W − /p,p ( I )-functions again lies in W − /p,p ( I ). Thus if µ ′ satisfies the conditions ofLemma 5.7, then the conclusion still holds for µ ′ ( t ) = µ ′ ( t )Π Nj =1 | t − t j | λ j where t , . . . t N ∈ I are distinct, and either λ j ≥ λ j > p − /p forall j , because the (normalized) outer function with modulus | t − t j | λ j / isjust ( t − t j ) λ j / , where the branch of the power λ j / dµ ( ξ ) = | − ξ | dm ( ξ ) provides us with an examplefor which (5.8) holds uniformly on T although µ ′ (1) = 0.5.3. Convergence of ORFs for Szeg˝o measures.
In this subsection,we assume as always that µ is a finite and positive measure with infinitesupport on T , but we no longer require it has unit mass. The ORFs and theassociated Carath´eodory and Szeg˝o functions are defined as before. Becausemultiplying µ by λ > φ n by λ − / and of S by λ / , the results below are invariant under such scalings.Observe that, similarly to the classical situation, the ORF φ n solves theextremal problem(5.11) max ξ n ∈L n , || ξ n || µ ≤ (cid:8) | a n,n | : ξ n = a n,n B n + a n,n − B n − + · · · + a n, B (cid:9) . We denote the value of the problem by κ n = κ n [ µ ], i.e. κ n = | φ ∗ n ( α n ) | .The extremal property (5.11) can also be recast as(5.12) κ − n = min ξ n ∈L n , ξ n ( α n )=1 k ξ n k µ , where the extremal value is uniquely attained at ξ n = φ ∗ n /φ ∗ n ( α n ). ULTIPOINT SCHUR APPROXIMATION 31
The Szeg˝o-type theorem we shall prove deals with the asymptotic behaviorof κ n as n → + ∞ , which entails further asymptotics for φ ∗ n .The statement is as follows. Theorem 5.8.
Let (0.3) , (0.11) - (0.13) be in force, and µ ∈ (S) . Then (5.13) lim n | φ ∗ n ( α n ) | | S ( α n ) | (1 − | α n | ) = 1 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then one can replacerelations (0.11) and (0.12) with (0.15) . The proof of Theorem 5.8 requires several steps. We first look at smoothmeasures.
Proposition 5.9.
Assume (0.3) holds and µ ∈ (S) is absolutely continuous.If there is an open neighborhood O of Acc ( α k ) ∩ T where µ ′ ≥ δ > with µ ′ ∈ W − /p,p ( I ) for each component I of O , p > , then (5.13) is valid.Proof. Since R n ( ., α n ) is the orthogonal projection of E n ( ., α n ) on P n ( dµ n ), R n ( ., α n ) is a polynomial of degree at most n and the minimummin p n ∈P n k E n ( ., α n ) − p n k L ( dµ n ) is attained exactly for p n = R n ( ., α n ). But k E n ( ., α n ) − p n k L ( dµ n ) = Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − α n t π n ( α n ) S ( α n ) − p n ( t ) S ( t ) π n ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dm ( t ) . Hence, the polynomial P n minimizing(5.14) min p n ∈P n (cid:13)(cid:13)(cid:13)(cid:13) − α n t − p n ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) provides us with R n ( ., α n ) through the relation R n ( ., α n ) = π n ( α n ) S ( α n ) P n . In view of (5.2), we write(5.15) | φ ∗ n ( α n ) | | S ( α n ) | (1 − | α n | ) = (cid:12)(cid:12)(cid:12)(cid:12) P n ( α n ) S ( α n ) π n − ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) . We also have for every polynomial p n (cid:13)(cid:13)(cid:13)(cid:13) − ¯ α n t − p n ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − p n ( t ) S ( t ) π n − ( t ) (cid:19) t − α n (cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) − p n ( α n ) S ( α n ) π n − ( α n ) (cid:19) t − α n + (cid:18) p n ( α n ) S ( α n ) π n − ( α n ) − p n ( t ) S ( t ) π n − ( t ) (cid:19) t − α n (cid:13)(cid:13)(cid:13)(cid:13) . Consequently, (cid:13)(cid:13)(cid:13)(cid:13) − ¯ α n t − p n ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) = (cid:12)(cid:12)(cid:12)(cid:12) − p n ( α n ) S ( α n ) π n − ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) − | α n | + (cid:13)(cid:13)(cid:13)(cid:13)(cid:18) p n ( α n ) S ( α n ) π n − ( α n ) − p n ( t ) S ( t ) π n − ( t ) (cid:19) t − α n (cid:13)(cid:13)(cid:13)(cid:13) . (5.16) Thus, if there is a sequence of polynomials ( p n ) satisfying(5.17) (cid:13)(cid:13)(cid:13)(cid:13) − ¯ α n t − p n ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) = o (cid:18) − | α n | (cid:19) , then we also have (see (5.14)) (cid:13)(cid:13)(cid:13)(cid:13) − ¯ α n t − P n ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) = o (cid:18) − | α n | (cid:19) , and by (5.16) lim n P n ( α n ) S ( α n ) π n − ( α n ) = 1 . In this case relation (5.15) gives us the desired limit (5.13).Note that µ ′ is bounded on each component I of O , since W − /p,p ( I )consists of continuous functions for p >
2. Hence S is continuous on O byLemma 5.7, and meets the assumptions of Lemma 5.4. Let K be a compactneighborhood of Acc ( α k ) included in O and R n ∈ L n be the sequence ofrational functions given by the lemma. Put R n = p n /π n . As k / (1 − ¯ α n t ) k = 1 / (1 − | α n | ), we get (cid:13)(cid:13)(cid:13)(cid:13) − ¯ α n t − p n − ( t ) S ( t ) π n ( t ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ sup t ∈ T \ K | − α n t | (cid:13)(cid:13)(cid:13)(cid:13) − p n − Sπ n − (cid:13)(cid:13)(cid:13)(cid:13) + 11 − | α n | (cid:13)(cid:13)(cid:13)(cid:13) − p n − Sπ n − (cid:13)(cid:13)(cid:13)(cid:13) L ∞ ( K ) . Since K is a neighborhood of Acc ( α k ), the above supremum is boundedand the first summand in the right-hand side of the equation goes to zeroas n → ∞ by the properties of R n . As to the second summand, it is o (1 / (1 − | α n | )) since R n − S converges to 1 uniformly on K . Therefore thesequence ( p n − ) satisfies (5.17) whence (5.13) holds. (cid:3) The assumption on µ ′ was only to ensure that S is continuous on O . Ifthis is known to be the case, it is not needed.The next proposition will sharpen Proposition 5.9 in that it allows µ tohave a singular part. We need a preparatory lemma. Lemma 5.10.
Assume (0.3) holds and let E ⊂ T , | E | = 0 , have an openneighborhood U in C such that U ∩
Acc ( α k ) = ∅ . Then, to every ε > , thereexists an integer n and R n ∈ L n such that(i) | R n | ≤ ε on T ,(ii) | − R n | ≤ ε on T \U ,(iii) | R n | ≤ ε on E ,Proof. Since E = supp µ s has Lebesgue measure zero, one can find g ∈ A ( D )such that g = 1 on E and | g | < D \ E , cf. [18, Ch. 3, Exercise 2]. Pick m so large that | g m | < ε/ D \ U . Since (0.3) holds and (1 − g m ) ∈ A ( D ),we can find n and R n ∈ L n such that | − g m − R n | < ε/ D . (cid:3) We also take note of the identity(5.18) F µ = S [ µ ] /S [ e µ ] , ULTIPOINT SCHUR APPROXIMATION 33 where e µ is the Herglotz measure of 1 /F µ , see (0.4). Indeed, since Carath´e-odory functions are outer, both sides of (5.18) are outer functions, positiveat 0, with equal modulus a.e. on T as can be readily computed from (0.5). Proposition 5.11.
Assumptions being as in Proposition 5.9, except that µ may now have a singular part satisfying (0.13) , we have that (5.13) holds.Proof. In view of Corollary 5.3, all we have to prove is that(5.19) lim inf n (1 − | α n | ) | S ( α n ) | κ n [ µ ] ≥ . Assume first that µ ′ ≥ δ ′ > V ⊃ supp µ s in T and thatthe restriction of µ ′ to each component I of V lies in W − /q,q ( I ) for some q >
4. Then S is continuous on V . Thanks to (0.13), we may require inaddition that V ∩ O = ∅ and also, by the compactness of supp µ s , that V has finitely many components. Fix a neighborhood W of supp µ s in T with W ⊂ V . We can apply Proposition 5.6 to dµ ac on each component I of V with K = W ∩ I and deduce from (5.8), since W remains at positive distancefrom ( α k ), that each ORF θ j associated with µ ac and with any subsequence( β l ) of ( α k ) is bounded by a constant C on W , where C is independent of j and of the subsequence. In particular, since µ ′ ∈ L ( T ) and | θ ∗ j | = | θ j | on T , to any ε > η > j ∈ N , ( β l ) ⊂ ( α k ),(5.20) Z W | θ ∗ j | µ ′ dm < ε, as soon as W ⊂ W has Lebesgue measure less than η .Pick ε > W ⊂ W be an open neighborhood of supp µ s in T such that |W | < η . This is possible since | supp µ s | = 0. Write W = U ∩ T where U is open in C and U ∩ ( α k ) = ∅ . This can be ensured because W ∩
Acc ( α k ) = ∅ . Apply Lemma 5.10 with E = supp µ s , and let R n ∈ L n be as in the lemma. Consider the sequence ( θ j ) of ORFs associated with dµ ac for the truncated sequence β l = α l + n , l ≥
1. Hence for n > n , wehave that | θ ∗ n − n ( α n ) | = κ ′ n − n [ µ ac ] where the prime in κ ′ n − n [ µ ac ] indicatesthat we work with the truncated sequence ( α k ) k>n . By (5.12), we get κ − n [ µ ] ≤ Z T (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n R n θ ∗ n − n ( α n ) R n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) dm + Z T (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n R n θ ∗ n − n ( α n ) R n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) dµ s . On the one hand, by properties ( ii ) and ( iii ) of Lemma 5.10, we get(5.21) Z T (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n R n θ ∗ n − n ( α n ) R n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) dµ s ≤ ε C (1 − ε ) ( κ ′ n − n [ µ ac ]) − . On the other hand, by properties ( i ) and ( ii ) of the same lemma, Z T (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n R n θ ∗ n − n ( α n ) R n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) dm ≤ (1 + ε ) (1 − ε ) Z T \W (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n θ ∗ n − n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) dm + (2 + ε ) (1 − ε ) Z W (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n θ ∗ n − n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) dm. Expanding (2+ ε ) and collecting terms, while using (5.20) and rememberingthat k θ ∗ n − n k µ ac = 1, we obtain Z T (cid:12)(cid:12)(cid:12)(cid:12) θ ∗ n − n R n θ ∗ n − n ( α n ) R n ( α n ) (cid:12)(cid:12)(cid:12)(cid:12) µ ′ ( t ) dm ≤ (1 + ε ) (1 − ε ) (cid:0) κ ′ n − n [ µ ac ] (cid:1) − + (3 + 2 ε ) ε (1 − ε ) (cid:0) κ ′ n − n [ µ ac ] (cid:1) − . (5.22)Since ε can be made arbitrarily small, we gather from (5.21) and (5.22) thatto each ε ′ > n such that κ − n [ µ ] ≤ (1 + ε ′ ) (cid:0) κ ′ n − n [ µ ac ] (cid:1) − as soon as n > n . But from Proposition 5.9 we know thatlim inf n (1 − | α n | ) | S ( α n ) | (cid:0) κ ′ n − n [ µ ac ] (cid:1) ≥ , so we obtain (5.19) since ε ′ is arbitrarily small.Next, we remove the assumption that µ ′ ≥ δ ′ > V with µ ′ ∈ W − /q,q on its components, but we suppose that µ ′ < C < ∞ on V . Fix a neighbor-hood W of supp µ s such that W ⊂ V . To each η >
0, pick a neighborhood V η ⊂ W of supp µ s satisfying |V η | < η . Put dµ η = µ ′ η dm + dµ s , where µ ′ η ( t ) = µ ′ ( t ) for t
6∈ V η , µ ′ η = C on V η . Being a positive constant on V η , µ ′ η certainly meets the assumptions of the preceding part of the proof, so (5.19)holds for µ η . Clearly, from (5.11), κ n [ µ η ] ≤ κ n [ µ ] because µ ≤ µ η hence(5.23) lim inf n (1 − | α n | ) | S [ µ η ]( α n ) | κ n [ µ ] ≥ . Now, let V ′ be open in C and contain no α k , with V ′ ∩ T = V . Since S [ µ ]( z ) = S [ µ η ]( z ) exp Z V η t + zt − z log | µ ′ /C | dm ( t ) ! , we see by dominated convergence (remember log µ ′ ∈ L ( T )) that S [ µ ] /S [ µ η ]converges uniformly to 1 in D \ V ′ as η →
0. Consequently (5.19) followsfrom (5.23).We now address the case where µ ′ may be unbounded in the neighborhood V of supp µ s but µ ′ ≥ δ ′′ > p > S [ µ ] is continuous on O byLemma 5.7. Observe also that F µ ac = µ ′ + i ˇ µ ′ lies in W − /p,p ( I ) for eachcomponent I of O and that F µ s is smooth on I , hence F µ = F µ ac + F µ s iscontinuous and bounded on some open neighborhood N of Acc ( α k ) ∩ T with N ⊂ O . Moreover | F µ | > δ > O , thus S [ e µ ] is continuous and positivelybounded from below on N by (5.18). Similarly | F µ | ≥ δ ′′ a.e. on V , thus e µ ′ = Re 1 /F µ is bounded there. In addition, supp e µ s ∩ N = ∅ otherwisethe H ∞ ( D )-function e − /F µ would have a singular inner factor which is notanalytic across N , and its modulus could not be continuous and nonzeroon N whereas | F µ | ≥ δ there [18, Ch. II, Theorem 6.2]. Therefore we canapply the previous case of the proof to e µ with O replaced by N ; indeed,we know that S [ e µ ] is continuous on N which is enough to proceed by theremark after Proposition 5.9. Thus, we obtain for the ORFs of the secondkind ψ n [ µ ]:(5.24) lim n | ψ ∗ n [ µ ]( α n ) | | S [ e µ ]( α n ) | (1 − | α n | ) = 1 . ULTIPOINT SCHUR APPROXIMATION 35
Recalling from Proposition 2.4 that ψ ∗ [ µ ]( α n ) /φ ∗ [ µ ]( α n ) = F µ ( α n ), we con-clude in view of (5.24) and (5.18) that (5.13) again holds.Finally, under the sole assumptions of the proposition, let V be an openneighborhood of supp µ s in T such that V ∩ O = ∅ , and V ′ be open in C andcontain no α k , with V ′ ∩ T = V . Let W be another neighborhood of supp µ s with W ⊂ V . Put dµ ε = µ ′ ε dm + dµ s with µ ′ ε = µ ′ + ε on W , and µ ′ ε = µ ′ on T \W . By what we just provedlim inf n (1 − | α n | ) | S [ µ ε ]( α n ) | κ n [ µ ε ] ≥ , and since κ n [ µ ε ] ≤ κ n [ µ ] (because µ ε ≥ µ ) while S [ µ ] /S [ µ ε ] = exp (cid:18)Z W t + zt − z log | µ ′ / ( µ ′ + ε ) | dm ( t ) (cid:19) , z ∈ D , converges uniformly to 1 in D \ V ′ as ε → µ ′ / ( µ ′ + ε ) to 1 a.e. on W , we conclude that (5.19) holds. (cid:3) In the course of the previous proof, we noticed that (5.24) is equivalentto (5.13). This is worth recording, taking into account that ee µ = µ : Corollary 5.12.
Let µ ∈ (S) . Then (5.13) holds for µ if, and only if itholds for e µ , the Herglotz measure of /F µ (see (0.4) ). There are Carath´eodory functions, with continuous and strictly positivereal part on T , whose imaginary part is unbounded. One example is 2 + ϕ where ϕ conformally maps D onto { z = x + iy ; | x | < / (1 + y ) } , ϕ (0) = 0and ϕ ′ (0) >
0, whose imaginary part is unbounded at ± i , see [18, Ch. III,Sect. 1]. If we put dµ ′ ( t ) = (2 + Re ϕ ( t )) dt , then 2 + ϕ ( t ) = F µ and and e µ ′ = µ ′ / | F µ | is continuous but vanishes at ± i . Letting ( α k ) accumulateat ± i , Theorem 5.8 will apply to µ and then Corollary 5.12 will provide uswith an example where (5.13) holds although (0.12) fails. Proof of Theorem 5.8.
Let O be the neighborhood of Acc ( α k ) ∩ T granted by(0.11)-(0.13). Shrinking O if necessary, we may assume that µ ′ is continuouswith µ ′ ≥ δ > O in T . Pick ε > < r < h r ( z ) = P rz ∗ µ ′ satisfies | h r − µ ′ | < ε on O .Let µ ε have singular part µ s and absolutely continuous part µ ′ ε dm where µ ′ ε ( t ) = µ ′ ( t ) for t / ∈ O and µ ′ ε ( t ) = h r ( t )+ ε for t ∈ O . Then µ ′ ≤ µ ′ ε ≤ µ ′ +2 ε on T and µ ′ ε is smooth on O . By Proposition 5.11, we have(5.25) lim n κ n [ µ ε ] | S [ µ ε ]( α n ) | (1 − | α n | ) = 1 . Since κ n [ µ ] ≥ κ n [ µ ε ] because µ ≤ µ ε , we deduce from (5.25) thatlim inf n (1 − | α n | ) | S ( α n ) | κ n [ µ ] ≥ lim inf n | S ( α n ) | | S [ µ ε ]( α n ) | (1 − | α n | ) | S [ µ ε ]( α n ) | κ n [ µ ε ](5.26) = lim inf n | S ( α n ) | | S [ µ ε ]( α n ) | . Recalling the inequalities on µ ′ , µ ′ ε given above, we get(5.27) | S ( z ) || S [ µ ε ]( z ) | = exp (cid:0) P z ∗ log( µ ′ /µ ′ ε ) (cid:1) ≥ − ε/δ for z ∈ D , and letting ε → α k ) accumulatesnontangentially on Acc ( α k ) ∩ T and that (0.11), (0.12) get replaced by (0.15).Then we can find a sequence of continuous functions ϕ j > µ ′ on O . Letting dµ j = dµ s + µ ′ j dm where µ ′ j = ϕ j on O and µ ′ j = µ ′ on T \ O , we get from the first part of the proof that (5.13) holdsfor µ j . Since µ j ≥ µ , we deduce as in (5.26) that for each j (5.28) lim inf n (1 − | α n | ) | S ( α n ) | κ n [ µ ] ≥ lim inf n | S ( α n ) | | S [ µ j ]( α n ) | . Without loss of generality, we may assume that ( α n ) converges to α ∈ D .If α ∈ D , the conclusion follows from the fact that S [ µ j ]( α ) → S ( α ) by themonotone convergence of µ ′ j to µ ′ . If α ∈ T , then by Fatou’s theoremlim n | S ( α n ) | | S [ µ j ]( α n ) | = µ ′ ( α ) µ ′ j ( α ) , which can be made arbitrarily close to 1 since lim j µ ′ j ( α ) = µ ′ ( α ) > (cid:3) Corollary 5.13.
Let (0.3) , (0.11) - (0.13) be satisfied and µ ∈ (S) . Then (5.29) lim n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Sφ ∗ n ( z ) − β n p − | α n | − α n z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 , where the unimodular factors β n are defined in Theorem 3’ of Section 0.2.Moreover, for any sequence ( z n ) ⊂ D , it holds that (5.30) lim n ( φ ∗ n ( z n ) S ( z n ) p − | z n | − β n p − | α n | p − | z n | − α n z n ) = 0 . If ( α k ) accumulates nontangentially on Acc ( α k ) ∩ T , then it is enough toassume instead of (0.11) - (0.12) that (0.15) holds.Proof. Estimating the integral, we get (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Sφ ∗ n ( z ) − β n p − | α n | − α n z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = Z T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Sφ ∗ n ( z ) − β n p − | α n | − α n z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dm ( z ) ≤ k φ n k µ − β n iπ Z T Sφ ∗ n ( z ) p − | α n | z − α n dz ! + 1= 2(1 − p − | α n | | S ( α n ) | | φ ∗ n ( α n ) | ) , and Theorem 5.8 yields (5.29). Next, let us set k z n = p − | z n | − zz n , G n ( z ) = Sφ ∗ n ( z ) − β n p − | α n | − α n z . Since k k z n k = 1, the relation just proven and the Schwarz inequality yieldlim n ( G n , k z n ) = 0. Expanding the scalar product gives us (5.30). (cid:3) Under the assumptions of the theorem, its conclusions also hold for ˜ µ byCorollary 5.12. ULTIPOINT SCHUR APPROXIMATION 37
Corollary 5.14.
Let (0.3) hold and µ ∈ (S) meet (0.11) - (0.13) . If I is anopen arc on T , with I ∩ supp µ s = ∅ , such that µ ′ ∈ W − /p,p ( I ) for some p > , then F µ φ ∗ n − ψ ∗ n converges to zero locally uniformly on I .Proof. From Corollary 5.13, limit (5.29) holds as well as its analogue for e µ :(5.31) lim n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) S [ e µ ] ψ ∗ n ( z ) − β n [ e µ ] p − | α n | − α n z (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = 0 . Moreover, it follows immediately from (5.18) and Proposition 2.4 that β n [ µ ] = β n [ e µ ]. Substracting (5.31) from (5.29) and using (5.18) now gives uslim n k S [ e µ ]( F µ φ ∗ n − ψ ∗ n ) k = lim n k S [ µ ] φ ∗ n − S [ e µ ] ψ ∗ n k = 0 . In particular, we get that from any subsequence of g n := F µ φ ∗ n − ψ ∗ n onecan extract a subsequence that converges pointwise a.e. to zero on T . Butsince g n is equicontinuous on compact subsets of I by Proposition 5.6, ii ),we deduce from Ascoli’s theorem that g n converges locally uniformly to zeroon I . (cid:3) Acknowledgments.
The second author is grateful to members of the APICSteam from INRIA Sophia-Antipolis for numerous invitations and warm hos-pitality.
References [1] Adams, R., Fournier J. Sobolev spaces. Pure and Applied Maths., vol. 140, AcademicPress, 2003.[2] Akhiezer, N. Theory of approximation. Dover Publications Inc., New York, 1992.[3] Anderson, B., Vongpanitlerd, S. Network Analysis and Synthesis - A modern systemtheory approach. Prentice Hall, 1973.[4] Aptekarev, A. Sharp constant for rational approximation of analytic functions. Math.Sb. 193 (2002), no. 1-2, 1–72.[5] Aptekarev A., Van Assche, W. Scalar and Matrix Riemann-Hilbert approach to thestrong asymptotics of Pad´e approximants and complex orthogonal polynomials withvarying weight. J. Approx. Theory 129 (2004), 129–166.[6] Astala K., Iwaniec T., Martin G. Elliptic partial differential equations and quasicon-formal mappings in the plane. Math. Series 48, Princeton Univ. Press, 2009.[7] Baratchart, L., K¨ustner, R., Totik, V. Zero distribution via orthogonality. Ann. Inst.Fourier 55 (2005), 1455–1499.[8] Baratchart, L., Yattselev, M. Convergent interpolation to Cauchy integrals over an-alytic arcs. to appear in Found. Constr. Math.[9] Begehr, H. Complex analytic methods for partial differential equations. World Scien-tific, 1994.[10] Bultheel, A., Gonz´alez-Vera, P., Hendriksen, E., Nj˚astad, O. Orthogonal rationalfunctions. Cambridge University Press, Cambridge, 1999.[11] Bultheel, A., Gonz´alez-Vera, P., Hendriksen, E., Nj˚astad, O. Orthogonal rationalfunctions on the unit circle: from the scalar to the matrix case. Lecture Notes inMath., vol. 1883 (2006), 187–228.[12] Campanato, S. Elliptic systems in divergence form, Interior regularity, Quaderni,Scuola Normale Superiore Pisa, 1980.[13] Dzrbasjan, M. Orthogonal systems of rational functions on the unit circle with givenset of poles. Dokl. Akad. Nauk. SSSR 147 (1962), 1278–1281 (Russian); English transl.in Soviet Mat. Dokl., (1962) no. 3, 1794–1798.[14] Denisov, S., Kupin, S. Orthogonal polynomials and a generalized Szeg˝o condition. C.R. Math. Acad. Sci. Paris 339 (2004), no. 4, 241–244. [15] Denisov, S., Kupin, S. Asymptotics of the orthogonal polynomials for the Szeg˝o classwith a polynomial weight. J. Approx. Theory 139 (2006), no. 1-2, 8–28.[16] Duren, P. Theory of H p spaces. Academic Press, New York, 1970.[17] Faurre, P., Clerget, M., Germain, F. Operateurs rationels positifs. Dunod, 1979.[18] Garnett, J. Bounded analytic functions. Springer, New York, 2007.[19] Geronimus, J. On polynomials orthogonal on the circle, on trigonometric moment-problem and on allied Carath´eodory-Schur functions. Rec. Math. [Mat. Sbornik] N.S. 15 (1944), no. 57, 99–130.[20] Geronimus, J. Orthogonal polynomials. Consultants Bureau, New York, 1961.[21] Grisvard, P. Elliptic problems in nonsmooth domains. Monographs and Studies inMathematics
24, Pitman, 1985.[22] Khrushchev, S. Schurs algorithm, orthogonal polynomials, and convergence of Wall’scontinued fractions in L ( T ). J. Approx. Theory 108 (2001), no. 2, 161–248.[23] Khrushchev, S. Classification theorems for general orthogonal polynomials on theunit circle. J. Approx. Theory 116 (2002), no. 2, 268–342.[24] Killip, R., Simon, B. Sum rules for Jacobi matrices and their applications to spectraltheory. Ann. of Math. (2) 158 (2003), no. 1, 253–321.[25] Koosis, P. Introduction to H p spaces. Cambridge University Press, Cambridge, 1998.[26] Krein, M. On a generalization of some investigations of G. Szeg˝o, V. Smirnoff and A.Kolmogoroff. C. R. (Doklady) Acad. Sci. URSS (N.S.) 46 (1945). 91–94.[27] Kuijlaars, A., McLaughlin, K., Van Assche, W., Vanlessen, M. The Riemann-Hilbertapproach to strong asymptotics for orthogonal polynomials on [ − , ∂ steepest descent method for orthogonal polynomialson the real line with varying weight, submitted.[32] Mart´ınez-Finkelstein, A., McLaughlin, K., Saff, E. Asymptotics of orthogonal poly-nomials with respect to an analytic weight with algebraic singularities on the circle.Int. Math. Res. Notices ID 91426 (2006), 1-43.[33] Matthaei, Y. Microwave filters, impedance matching networks and coupling struc-tures. New York, Mc Graw Hill, 1965.[34] Mi˜na-D`ıaz, E. An expansion for polynomials orthogonal over an analytic Jordancurve. Comm. Math. Phys. 285 (2009), no. 3, 1109–1128.[35] Nevanlinna, R. ¨Uber beschr¨ankte Funktionen, die in gegebenen Punktenvorgeschriebene Werte annehemen. Ann. Acad. Sci. Fenn. 13 (1919) no. 1.[36] Nikishin, E., Sorokin, V. Rational approximations and orthogonality. Translations ofAMS, vol. 92, Providence, RI, 1991.[37] Njastad, O., Vel`azquez, L. Wall rational functions and Khrushchev’s formula fororthogonal rational functions. Constr. Approx. 30 (2009), no. 2, 277-297.[38] Pan, K. On the convergence of rational functions orthogonal on the unit circle. J.Comput. Appl. Math. (1996), no. 76, 315–324.[39] Peherstorfer, F., Yuditskii, P. Asymptotics of orthonormal polynomials in the pres-ence of a denumerable set of mass points. Proc. Amer. Math. Soc. 129 (2001), no. 11,3213–3220.[40] Rakhmanov, E. Asymptotic properties of polynomials orthogonal on the unit circlewith weights not satisfying the Szeg˝o condition (Russian). Mat. Sb. 130 (172) (1986),no. 2, pp. 151-169, 284.[41] Remling, C. The absolutely continuous spectrum of Jacobi matrices, submitted.[42] Schur, I. ¨Uber potenzreihen, die im innern des einheitskreises beschr¨ankt sind. J.Reine Angew. Math. 147 (1917) 205232. English translation in: I. Schur methods inoperator theory and signal processing (Operator Theory: Adv. and Appl. 18 (1986),Birkh¨auser Verlag). ULTIPOINT SCHUR APPROXIMATION 39 [43] Simon, B. Orthogonal polynomials on the unit circle, I, II. AMS Colloquium Publi-cations, vol. 54, Providence, RI, 2005.[44] Szeg˝o, G. Orthogonal polynomials. AMS, Providence, RI, 1975.[45] Wall, H. Continued fractions and bounded analytic functions. Bull. Amer. Math. Soc.50 (1944). 110–119.[46] Wall, H. Analytic theory of continued fractions. Van Nostrand, New York, 1948.[47] Ziemer, W. P. Weakly differentiable functions. G.T.M. 120, Springer, 1989.
E-mail address : [email protected]
IMB, Universit´e Bordeaux 1, 351 cours de la Lib´eration, 33405 TalenceCedex France
E-mail address : [email protected] E-mail address : [email protected] E-mail address ::