On a new approach to the problem of the zero distribution of Hermite-Padé polynomials for a Nikishin system
OON A NEW APPROACH TO THE PROBLEM OF THEZERO DISTRIBUTION OF HERMITE–PAD´EPOLYNOMIALS FOR A NIKISHIN SYSTEM
SERGEY P. SUETIN
Abstract.
A new approach to the problem of the zero distributionof Hermite–Pad´e polynomials of type I for a pair of functions f , f forming a Nikishin system is discussed. Unlike the traditional vectorapproach, we give an answer in terms of a scalar equilibrium problemwith harmonic external field, which is posed on a two-sheeted Riemannsurface.Bibliography: [50] titles.Paper: http://mi.mathnet.ru/eng/tm3908 Keywords: Hermite–Pad´e polynomials, non-Hermitian orthogonal polynomials,distribution of the zeros
Contents
1. Introduction and statement of the problem 12. Proof of Theorem 1 83. Proof of Theorem 2 9References 191.
Introduction and statement of the problem f ( z ) := 1( z − / , f ( z ) := (cid:90) − h ( x )( z − x ) dx √ − x , z ∈ D := C \ E ;(1)here E := [ − , h is a holomorphic function on E (written h ∈ H ( E )) ofthe form h ( z ) = (cid:98) σ ( z ), where (cid:98) σ ( z ) := (cid:90) F dσ ( t ) z − t , z ∈ C \ F, F := p (cid:71) j =1 [ c j , d j ] ⊂ R \ E, (2) c j < d j , σ is a positive Borel measure with support in F and such that σ (cid:48) := dσ/dx > F . Functions (cid:98) σ ( z ) in (2) arecalled Markov functions. Regarding the choice of branches of the function( · ) / and of the root √ · in (1), see § Date : 24.11.2017.This research was carried out with the partial support of the Russian Foundation forBasic Research (grant no. 15-01-07531). a r X i v : . [ m a t h . C V ] M a y SERGEY P. SUETIN
For a tuple [1 , f , f ] of three functions, where f and f are given by (1),and an arbitrary n ∈ N , Hermite–Pad´e polynomials of type I Q n, , Q n, , Q n, ,deg Q n,j (cid:54) n , Q n,j (cid:54)≡
0, of order n are defined (not uniquely) from therelation R n ( z ) := ( Q n, · Q n, f + Q n, f )( z ) = O (cid:18) z n +2 (cid:19) , z → ∞ . (3)The purpose of the present paper is to put forward and discuss, on anexample of a pair of functions of the form (1), a new approach to the studyof the limit distribution of the zeros for Hermite–Pad´e polynomials of type Ias defined by (3). As it is our intention to apply, in subsequent studies,this approach to fairly general classes of analytic functions (see the resultannounced in [50] and Remark 1 below), we shall first give the notation tobe used below (in this respect, see [46], [32], [49]).Let Σ ⊂ C be an arbitrary finite set, card Σ < ∞ . We let A ◦ (Σ) denotethe class of all analytic functions which are holomorphic at each point z ∈ C \ Σ, admit analytic continuation from z along any path γ in C disjointfrom Σ, and such that at least one point of the set Σ is a branch point ofthis function. For f , f ∈ A ◦ (Σ) (under the assumption that the functions1 , f , f are independent over the field C ( z ) of rational functions of z withcomplex coefficients), the problem of the limit distribution of the zeros ofHermite–Pad´e polynomials has a long history and in general is still unsolved(see [36], [43], [3], [41]). There is also no complete understanding what termsshould be employed to solve this problem. At present, the answer to theproblem of the limit distribution of the zeros of Hermite–Pad´e polynomialsis available only for some particular classes of analytic functions (see [17],[34], [38], [19], [2], [4], [40], [32]). As a rule, the limit distribution of the zerosof Hermite–Pad´e polynomials for a pair of functions f , f can be describedfollowing the approach first proposed by Nuttall (see [36], [38]) in termsrelated to some three-sheeted Riemann surface which in a certain sense is “associated” with the pair of functions f , f (for the relation betweenthe three-sheeted Riemann surface with the asymptotics of Hermite–Pad´epolynomials, see also [25], [6], [26].)For a pair of functions f , f of form (1) the above problem was solved byNikishin [34] in 1986 (see also [33], [35], [7]). Note that in [34] the problemwas solved for an arbitrary number of functions f , f , . . . , f m forming a Nik-ishin system ; a pair of functions (1) is a particular case of such a system.The solution of the problem of the distribution of the zeros of Hermite–Pad´epolynomials in [34] is based on the potential theory approach developed byGonchar and Rakhmanov [17] in 1981 for the purposes of solving the zerodistribution problem for Hermite–Pad´e polynomials of type II forming an
Angelesco system (a particular case of an arbitrary number of functions f , f , . . . , f m was also considered in the paper [17], in which, in particular,the effect of pushing of the support of the equilibrium measure inside theoriginal orthogonality interval was discovered; see also [42]). Within theframework of this vector approach, the answer for a pair of functions (1) is Similarly to the way the strong asymptotics of Pad´e polynomials is described in termsrelated to the two-sheeted Riemann surface associated (in accordance with the Stahl the-ory) with an arbitrary function from the class A ◦ (Σ); see [37], [5], [31]. ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 3 given in terms of a vector-equilibrium measure (cid:126)λ = ( λ , λ ) supported onthe vector-compact set ( E, F ) (that is, supp λ ⊂ E, supp λ ⊂ F ). Theequilibrium conditions are determined by the interaction matrix of mea-sures M Nik = (cid:18) − − (cid:19) , which is known as the Nikishin matrix . Thesolution of the problem is a unique vector-measure (cid:126)λ = ( λ , λ ) with sup-port on the vector-compact set ( E, F ); this measure is extremal for the en-ergy functional defined by the logarithmic kernel and the interaction matrix M Nik (see [35], [19], [3], [28]). The extremal vector-measure (cid:126)λ = ( λ , λ ) iscompletely characterized by the equilibrium condition for the correspondingvector potential and the vector-compact set ( E, F ); for more details, see[3], [28].Note that, for arbitrary functions f , f ∈ A ◦ (Σ), the problem of thelimit distribution of the zeros of the corresponding Hermite–Pad´e polyno-mials turns out to be equivalent to the problem of the limit distribution ofthe zeros of polynomials satisfying some non-Hermitian orthogonality con-ditions (see [18], [39], [41], [42]). The characteristic feature of non-Hermitianorthogonality conditions is that the contour of integration is not fixed a pri-ori , but rather lies in some class of “admissible” contours. The followingheuristic conclusion can be made based on a series of particular cases in-vestigated so far: in this class, there exists a unique “optimal” contour attracting in the limit the zeros of Hermite–Pad´e polynomials. The “opti-mality” property of a contour is formulated in terms of the correspondingvector equilibrium problems of potential theory. This optimal contour pos-sesses a certain vector S -property, which completely characterizes it in theclass of admissible vector-contours. In modern terms, such a contour iscalled an S -curve or an S -compact set (see [39]).The concept of an S -compact set was first introduced by H. Stahl in the1985–1986s (see [44] and [45] and there references given therein) when con-sidering the problem of the limit distribution of the zeros and poles of Pad´eapproximants in the class of multivalued analytic functions A ◦ (Σ). In 1987Gonchar and Rakhmanov [18], in their solution of the “1 / external field ”defined by a harmonic function (more general external fields and the corre-sponding S -curves were considered in [39]). This new approach was used in2012–2015 by Buslaev [9]–[11] to solve the problem of the limit distributionof the zeros and poles of multivalued Pad´e approximants. Here, the potentialof a negative unit charge concentrated at a finite number of interpolationnodes appears naturally as an external field (see also [13], [15], [14]).The class of methods developed by H. Stahl, A. A. Gonchar, and E. A.Rakhmanov in the 1980s for the purpose of studying the limit distribution ofthe zeros of non-Hermitian orthogonal polynomials is called at present the Here and below, by a contour we shall mean a composite contour consisting of a finitenumber of closed curves and splitting the Riemann sphere into a finite number of domains;see [10], [11].
SERGEY P. SUETIN
Gonchar–Rakhmanov–Stahl method (or briefly the GRS- method ); see [46],[32], [41], [42].The purpose of the present paper is, by using an example of two functions f and f of the form (1), put forward and discuss a new approach to theproblem of the limit distribution of the zeros of Hermite–Pad´e polynomials,which in a certain sense further develops the approach of A. A. Goncharand E. A. Rakhmanov employed in their solution of the “1 / Q n, as n →∞ will be characterized in terms related to some scalar potential theoryequilibrium problem (but with external field), which in addition is posed noton the Riemann sphere C , but rather on the two-sheeted Riemann surfaceof the function w = z −
1. This is the principal distinguishing feature ofthe approach of the present paper from the standard method based on thevector equilibrium problem posed on the Riemann sphere.Let us clarify the choice of the pair of functions (1) to illustrate the newapproach and the fact that here we speak only about the distribution of thezeros of the polynomial Q n, .The thing is, on the one hand, as we have already mentioned, in theclass A ◦ (Σ) the problem of the distribution of the zeros of Hermite–Pad´epolynomials for an arbitrary pair of independent functions f , f ∈ A ◦ (Σ)is not yet solved and it is even unclear what terms should be employed tofind its solution (for conjectures in this direction, see [36], [3], [43], [41]). Inparticular, there is no solution in this problem even for a pair of functionswith two branch points, of which each is in “the general position”. On theother hand, for the Pad´e polynomials P n, , P n, , deg P n,j (cid:54) n , P n,j (cid:54)≡
0, asdefined from the relations( P n, + P n, f )( z ) = O (cid:18) z n +1 (cid:19) , z → ∞ , (4)where f ∈ H ( ∞ ), Stahl’s theory is valid for an arbitrary function f fromthe class A ◦ (Σ). This leads to the following fairly natural argument: one ofthe functions, f say, in the relation (3) defining the Hermite–Pad´e polyno-mials, should be taken as simple as possible (retaining the independence oftwo functions f , f ) with the aim at maximally extending the calculus toa larger class of functions that contains the second function f . In this way,using the approach proposed here, we managed to substantially enlarge theclass of functions containing the function h in representation (1). Namely,for an arbitrary function h ∈ A ◦ (Σ), where Σ ⊂ C \ E , it is possible tocharacterize completely the problem of the limit distribution of the zeros ofthe polynomials Q n, in terms of the same scalar potential theory equilib-rium problem with an external field. This problem is posed on the sametwo-sheeted Riemann surface of the function w = z − S -compactset F corresponding to the problem under consideration and which replacesthe union of a finite number of closed intervals (see (2)). The corresponding Stahl’s theory is much more general and can be applied to any multivalued analyticfunction, whose singular set is of zero logarithmic capacity.
ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 5 result was announced in [50]; the author intends to give the proof of thisresult in a separate paper.We note the papers [40], [46] and [24], in which the equilibrium prob-lem for a mixed Green-logarithmic potential was employed for the study ofthe limit distribution of the zeros of Hermite–Pad´e polynomials for a tu-ple [1 , f , f ], where a pair of functions f , f forms a generalized (complex)Nikishin system (see also [12], [47], [32], [41]). The method of investigationproposed in the present paper is different from that of [40], [46] and [24].Some precursor considerations and results that eventually culminated inthe statement of the potential theory equilibrium problem on the Riemannsurface w = z − f ( z ) := 1( z − / , f ( z ) := 1 (cid:0) ( z − . − . i )( z + . − . i ) (cid:1) / , forming an Angelesco system.The author is grateful to the referee for the many helpful comments andsuggestions which led to a great improvement in the presentation of thepaper and for calling his attention to the papers [1] and [29].1.2. We shall require the following notation and definitions. We set D := C \ E , ϕ ( z ) := z + ( z − / , z ∈ D, (5)where we choose the branch of the root function such that ( z − / /z → z → ∞ . For x ∈ ( − , √ − x we shall understand the positivesquare root: √ b = b for b (cid:62) Q ∈ C [ z ], Q (cid:54)≡
0, by χ ( Q ) := (cid:88) ζ : Q ( ζ )=0 δ ζ we shall mean the counting measure of the zeros of the polynomial Q (count-ing multiplicities). In what follows, given an arbitrary n ∈ N , we denote by P n := C n [ z ] the class of all algebraic polynomials of degree (cid:54) n with complexcoefficients. SERGEY P. SUETIN
We let R denote the two-sheeted Riemann surface of the function w = z − C with branch points at z = ±
1. Each (open) sheet of the Riemann surface R is the Riemann sphere cut along the interval E , the opposite sides ofcuts from different sheets being identified. The first (open) sheet R (1) of theRiemann surface R is that on which w = ( z − / ∼ z as z → ∞ ; onthe second sheet R (2) w = − ( z − / ∼ − z as z → ∞ . A point z on theRiemann surface R is the pair ( z, w ) = z ∈ R . The canonical projection π , π : R → C , is defined in the standard way: π ( z ) = z . Note that R isa Riemann surface of zero genus, and so any divisor d of degree 0 on R isa principal one; that is, there exists a meromorphic function on R whosedivisor of the zeros and poles coincides with d . From a given divisor ofdegree 0 such a meromorphic function is defined uniquely up to a nontriv-ial multiplicative constant (for a more detailed account of these and otheraspects of Riemann surfaces, see [16]).Thus, the function z + w , which is meromorphic on the Riemann surface R , will be denoted by Φ( z ) := z + w . Points of the Riemann surface lying onthe first (open) sheet R (1) will be denoted by z (1) ; by z (2) we denote pointsfrom the second sheet R (2) . So, z (1) = ( z, ( z − / ), z (2) = ( z, − ( z − / ), π ( R (1) ) = π ( R (2) ) = D .The following identity is easily verified for z , a ∈ R \ Γ z − a ≡ − [Φ( z ) − Φ( a )][1 − Φ( z )Φ( a )]2Φ( z )Φ( a ) . (6)Indeed, each of the functions on the right and left of (6) is meromorphicon the Riemann surface R . The divisor z − a of the left-hand side can beeasily evaluated to be equal to d = −∞ (1) − ∞ (2) + a (1) + a (2) . For thedivisor of the right-hand side, we also have d = −∞ (1) − ∞ (2) + a (1) + a (2) .Hence, these two functions are identically equal except for a multiplicativeconstant, which can be easily calculated.From (6) we have, in particular, the identity z − a ≡ − [ ϕ ( z ) − ϕ ( a )][1 − ϕ ( z ) ϕ ( a )]2 ϕ ( z ) ϕ ( a ) , (7)which holds for z, a ∈ D . The following identityΦ( z (1) )Φ( z (2) ) ≡ , z ∈ D, (8)can also be easily verified.Let M ( F ) be the space of all unit positive Borel measures supported ona compact set F . Given an arbitrary measure µ ∈ M ( F ), we define by V µ ( z ) := (cid:90) F log 1 | z − t | dµ ( t ) (9)the logarithmic potential of µ , I ( µ ) := (cid:90) (cid:90) F × F log 1 | z − t | dµ ( z ) dµ ( t ) = (cid:90) F V µ ( z ) dµ ( z ) (10) This identity, for all its undoubted simplicity, was first used very effectively in [20],formulas (15), (67).
ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 7 is the corresponding energy functional. By M ◦ ( F ) ⊂ M ( F ) we shall denotethe space of measures with finite energy, I ( µ ) < ∞ . We recall the positivityproperty of logarithmic energy with respect to neutral charges: I ( µ − ν ) (cid:62) ∀ µ, ν ∈ M ◦ ( F ) and I ( µ − ν ) = 0 ⇔ µ = ν. (11)For an account of these and other properties of logarithmic potentials em-ployed in the present paper, see [27].For a measure µ ∈ M ( F ), we set P µ ( z ) := (cid:90) F log | − ϕ ( z ) ϕ ( t ) || z − t | dµ ( t ) , ψ ( z ) := log | ϕ ( z ) | , (12)and define J ( µ ) : = (cid:90) (cid:90) F × F log | − ϕ ( z ) ϕ ( t ) || z − t | dµ ( z ) dµ ( t )= (cid:90) F P µ ( z ) dµ ( z ) ,J ψ ( µ ) : = (cid:90) (cid:90) F × F (cid:26) log | − ϕ ( z ) ϕ ( t ) || z − t | + ψ ( z ) + ψ ( t ) (cid:27) dµ ( z ) dµ ( t )= (cid:90) F P µ ( z ) dµ ( z ) + 2 (cid:90) F ψ ( z ) dµ ( z ) . (13)From identity (7) we have the following equality, which holds for z, ζ ∈ D ,log | − ϕ ( z ) ϕ ( ζ ) || z − ζ | = log 1 | z − ζ | + log 1 | ϕ ( z ) − ϕ ( ζ ) | + log 2 + ψ ( z ) + ψ ( ζ ) . (14)Potentials with kernels of the formlog 1 | z − ζ | + log 1 | v ( z ) − v ( ζ ) | , where z, ζ ∈ [ A, B ] ⊂ R , v ( z ) is an arbitrary nondecreasing function on[ A, B ], were considered in the paper [30], however, the author of the presentpaper is unaware of any applications of such potentials in the theory ofHermite–Pad´e polynomials.1.3. The main results of the present paper are Theorems 1 and 2.
Theorem 1.
In the class M ◦ ( F ) , there exists a unique measure λ = λ F ∈ M ◦ ( F ) such that J ψ ( λ ) = min µ ∈ M ( F ) J ψ ( µ ) . (15) The measure λ is completely characterized by the following equilibrium con-dition: P λ ( z ) + ψ ( z ) ≡ w F , z ∈ S ( λ ) , (cid:62) w F , z ∈ F \ S ( λ ) . (16) Theorem 2.
Let f and f be functions given by the representations (1) and let Q n, be the Hermite–Pad´e polynomial defined by (3) . Then n χ ( Q n, ) → λ, n → ∞ . (17) More precisely, the positivity of the logarithmic kernel. It is clear that ψ ( z ) = log ϕ ( z ) for z ∈ R \ E . SERGEY P. SUETIN
The convergence in (17) shall be understood in the sense of weak conver-gence in the space of measures. It may be pointed out once more that theassertion of Theorem 2 on the existence of the limit distribution of the zerosof the polynomials Q n, is not new (see, first of all, [34], and also [19], [4]).The new point here is the characterization of this limit distribution in termsthe scalar equilibrium problem (15)–(16). This was achieved by posing thecorresponding potential theory problem not on the Riemann sphere, but onthe two-sheeted Riemann surface of the function w = z − Proof of Theorem 1 U ⊃ F be some neighborhood of the compact set F such that U ∩ E = ∅ . For all µ ∈ M ( F ), the function (cid:90) F log | − ϕ ( z ) ϕ ( t ) | dµ ( t )is harmonic in U and the potential V µ ( z ) is a superharmonic function in U ,and hence, since cap F > M ( F ) is compact in the weak topology, andusing the principle of descent for logarithmic potentials (see [27], Ch. I, § λ ∈ M ◦ ( F ) satisfyingequality (15). Using identity (14), one can easily prove the convexity of theenergy functional J ψ ( · ), J ψ (cid:18) µ + ν (cid:19) (cid:54) (cid:2) J ψ ( µ ) + J ψ ( ν ) (cid:3) ∀ µ, ν ∈ M ( F ); (18)moreover, J ( µ − ν ) = 2 J ψ ( µ ) + 2 J ψ ( ν ) − J ψ (cid:18) µ + ν (cid:19) , (19) J ( µ − ν ) = (cid:90) (cid:90) F × F (cid:26) log 1 | z − ζ | + log 1 | ϕ ( z ) − ϕ ( ζ ) | (cid:27) d ( µ − ν )( z ) d ( µ − ν )( ζ ) , (20)for all µ, ν ∈ M ◦ ( F ). As a direct corollary of (18)–(20) we see that thefunctional J ( · ) is positive on neutral charges (cf. (11)), J ( µ − ν ) (cid:62) ∀ µ, ν ∈ M ◦ ( F ) and J ( µ − ν ) = 0 ⇔ µ = ν. Furthermore, the following equalities are easily verified: J ( µ ) = (cid:90) (cid:90) F × F (cid:26) log 1 | z − ζ | + log 1 | ϕ ( z ) − ϕ ( ζ ) | (cid:27) + log 2 + 2 (cid:90) ϕ ( z ) dµ ( z ) ,J ψ ( µ ) = (cid:90) (cid:90) F × F (cid:26) log 1 | z − ζ | + log 1 | ϕ ( z ) − ϕ ( ζ ) | (cid:27) + log 2 + 4 (cid:90) ϕ ( z ) dµ ( z ) . (21)2.2. Arguing as in Lemma 6 of [18], we can now prove the equilibriumproperty (16) of the extremal measure λ by using the above equalities andthe positivity property of the functional J ( · ) ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 9
Indeed one verifies directly that J ψ ( εν + (1 − ε ) λ ) − J ψ ( λ ) = 2 ε (cid:90) F ( P λ + ψ )( z ) d ( ν − λ ) + ε J ( ν − λ ) (22)for any ε > ν ∈ M ◦ ( F ). It follows that the minimizingmeasure λ is the only measure from M ◦ ( F ) satisfying the condition (cid:90) F ( P λ + ψ ) d ( ν − λ ) (cid:62) ∀ ν ∈ M ◦ ( F ) . (23)In the actual fact, (23) is an immediate consequence of (18), (11) and (22)as ε →
0. On the other hand, since the energy functional J ( · ) is positiveon neutral charges, we have J ( ν − λ ) (cid:62) ν ∈ M ◦ ( F ). Anappeal to (22) with ε = 1 shows that any measure ν ∈ M ◦ ( F ) satisfying (23)minimizes the energy integral J ψ ( · ). If a measure λ satisfies condition (23),then it obeys the equilibrium relations (16) with w F := (cid:90) F ( P λ + ψ ) dλ. Indeed, if P λ ( x ) + ψ ( x ) < w F on a closed set e ⊂ F , cap( e ) >
0, thenthere exists ν ∈ M ◦ ( e ), for which (cid:90) F ( P λ + ψ )( x ) dν ( x ) < w F , which showsthat (23) is violated. Hence ( P λ + ψ )( x ) (cid:62) w F everywhere on the (regular)compact set F . If ( P λ + ψ )( x ) > w F on a nonempty set e ⊂ S ( µ ), then theinequality (cid:90) ( P λ + ψ )( x ) dλ ( x ) > w F is secured by the lower semi-continuityof the function ( P λ + ψ )( z ), contradicting the definition of w F .If λ is an equilibrium measure, then P λ + ψ (cid:54) w F everywhere on S ( λ ),which shows that λ ∈ M ◦ ( F ). Since the sets of zero inner capacity playno role in integration with respect to measures in M ◦ ( F ), we obtain (23).Finally, ( P λ + ψ )( z ) ≡ w F on S ( λ ), because F is a regular compact set.Thus, the extremal measure λ , and only this measure, satisfies the equi-librium conditions (16). This proves Theorem 1.Note that F is a regular compact set, and hence the equilibrium measureis characterized by the equalitymin z ∈ F ( P λ + ψ )( z ) = max µ ∈ M ( F ) min z ∈ F ( P µ + ψ )( z ) . Proof of Theorem 2 (cid:90) γ ( Q n, + Q n, f + Q n, f )( z ) q ( z ) dz = (cid:90) γ ( Q n, f + Q n, f )( z ) q ( z ) dz, (24)which holds for any polynomial q ∈ P n ; in (24) γ is an arbitrary contourseparating the interval E from the infinity point z = ∞ . For completeness of presentation, we give the proof of (16), cf. Lemma 6 of [18].
Let P n, and P n, be the Pad´e polynomials for the function f ; that is,deg P n,j (cid:54) n , P n,j (cid:54)≡
0, and H n ( z ) := ( P n, + P n, f )( z ) = O (cid:18) z n +1 (cid:19) , z → ∞ . (25)It is known that P n, = T n are Chebyshev polynomials of the first kind thatare orthogonal on the interval E with the weight 1 / √ − x , H n is the corre-sponding function of the second kind. We shall assume that the Chebyshevpolynomials are normalized as follows: T n ( z ) = 2 n z n + · · · . Hence, for thefunctions of the second kind H n , we have H n ( z ) = κ n ϕ (cid:48) ( z ) ϕ n +1 ( z ) , κ n (cid:54) = 0 , H n ( z ) = 12 πi (cid:90) E T n ( x )∆ f ( x ) x − z dx, z ∈ D, (26)∆ H n ( x ) : = H n ( x + i − H n ( x − i T n ( x )∆ f ( x ) = T n ( x ) 2 i √ − x , x ∈ ( − , . (27)Besides, the polynomials T n and the functions of the second kind H n satisfythe same second-order recurrence relation, but with different initial data y k = 2 zy k − − y k − , k = 1 , , . . . , (28)where one should put y − ≡ y ≡ T k and y − ≡ y = f ( z ) = 1 / ( z − / for the functions of the second kind H k . We have (cid:90) γ p ( z ) f ( z ) T n + j ( z ) dz = 0 , j = 1 , , . . . , n for any polynomial p ∈ P n , and so from (24) with q = T n +1 , . . . , T n itfollows that (cid:90) γ Q n, ( z ) f ( z ) T n + j ( z ) dz = 0 , j = 1 , , . . . , n. (29)Next, using (29) and the definition (1) of the function f , we have (cid:90) E Q n, ( x ) T n + j ( x ) 1 √ − x h ( x ) dx = 0 , j = 1 , . . . , n. (30)In view of (27), the above relation is equivalent to the relation (cid:90) γ Q n, ( z ) H n + j ( z ) h ( z ) dz = 0 , j = 1 , . . . , n, (31)where γ is an arbitrary contour separating the interval E from the compactset F . Since h ( z ) = (cid:98) σ ( z ), relation (31) can be easily written in the form (cid:90) F Q n, ( x ) H n + j ( x ) dσ ( x ) = 0 , j = 1 , . . . , n. (32)These orthogonality relations will play a key role in the subsequent analysisof the limit distribution of the zeros of the polynomials Q n, .Let N , 0 (cid:54) N (cid:54) n , be an arbitrary natural number. We shall assumewithout loss of generality that N = 2 m is an even number (the case of an In view of the representation H n ( z ) = ϕ (cid:48) ( z ) /ϕ n +1 ( z ), the orthogonality relations (32)are similar to those considered in [30]. ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 11 odd N is treated similarly). Given arbitrary complex numbers c , . . . , c N ∈ C , consider the sum N (cid:88) j =1 c j H n + j ( z ) . By using the recurrence relations (28), this sum can be easily written as N (cid:88) j =1 c j H n + j ( z ) = q m, ( z ) H n + m +1 ( z ) + q m, ( z ) H n + m ( z ) , (33)where q m, , q m, ∈ P m − are polynomials of degree (cid:54) m −
1. Since the con-stants c , . . . , c N in (33) are arbitrary, it is easily verified that the polynomi-als q m, and q m, can also be chosen arbitrarily. So, using (33), relations (32)can be written in the following equivalent form (cid:90) F Q n, ( x ) (cid:8) q m, ( x ) H n + m +1 ( x ) + q m, ( x ) H n + m ( x ) (cid:9) dσ ( x ) = 0 (34)with arbitrary polynomials q m, ∈ P m − and q m, ∈ P m − . Now, from (34)and the available properties of the functions of the second kind (see (26)),we have0 = (cid:90) F Q n, ( x ) (cid:26) q m, ( x ) H n + m +1 H n + m ( x ) + q m, ( x ) (cid:27) H n + m ( x ) dσ ( x )= (cid:90) F Q n, ( x ) (cid:26) q m, ( x ) κ n + m +1 κ n + m ϕ ( x ) + q m, ( x ) (cid:27) κ n + m ϕ (cid:48) ( x ) ϕ n + m +1 ( x ) dσ ( x ) . (35)Now, using the definition of the function Φ( z ) (see sec. 1.2), which ismeromorphic on the Riemann surface R , we get the following orthogonalityrelation (cid:90) F Q n, ( x ) (cid:110) q m, ( x )Φ( x (2) ) + q m, ( x ) (cid:111) ϕ (cid:48) ( x )Φ( x (2) ) n + m +1 dσ ( x ) = 0 , (36)which holds for any polynomials q m, , q m, ∈ P m − .3.2. We now set g n ( z ) := q m, ( z )Φ( z ) + q m, ( z ) , (37)where it is assumed that deg q m, = deg q m, = m −
1. Then, for the divisorof the function g n we havediv( g n ) = − m ∞ (1) − ( m − ∞ (2) + N − (cid:88) j =1 a N,j , (38)where, as is clear, the zeros a N,j of the function g n can be chosen arbitrarily,because the polynomials q m, , q m, are arbitrary. Next, the function g n ismeromorphic on R and the genus of the Riemann surface R is zero, andhence the function g n is completely defined by its divisor (38) (of the zerosand poles). As a result, from (38) we have the following explicit representa-tion for the function g n : g n ( z ) = C N · N − (cid:89) j =1 (cid:2) Φ( z ) − Φ( a N,j ) (cid:3) · Φ( z ) − m +1 , C N (cid:54) = 0 . (39) Indeed, it is easily checked that the divisor of the zeros and poles of theright-hand side of (39) coincides with that of (38). Below, in accordancewith (36), we shall need to consider only the case when all points a N,j lieon the second sheet of the Riemann surface R , a N,j = a (2) N,j ∈ R (2) . Moreprecisely, the zeros a N,j should be as follows: they should lie on the secondlist and be such that π ( a N,j ) ∈ (cid:98) F \ E , where (cid:98) F is the convex hull of F . Inthis case, it follows from (39) that g n ( z (2) )Φ( z (2) ) n + m +1 = C N · N − (cid:89) j =1 (cid:2) Φ( z (2) ) − Φ( a (2) N,j ) (cid:3) · Φ( z (2) ) n +2 . (40)We now consider the product g n ( z )Φ( z ) n + m +1 . Using identities (6) and (8),we write it as g N ( z )Φ( z ) n + m +1 = C N · N − (cid:89) j =1 (cid:2) Φ( z ) − Φ( a N,j ) (cid:3) · Φ( z ) − m +1 Φ( z ) n + m +1 = (cid:101) C N · N − (cid:89) j =1 z − a N,j − Φ( z )Φ( a N,j ) · Φ( z ) N + n +1 , (41)where (cid:101) C N (cid:54) = 0 and it is assumed that all a N,j (cid:54) = ∞ (1) , ∞ (2) . In accordancewith (36), we shall require representation (41) only in the case when z = z (2) and all a N,j = a (2) N,j . In this setting, we have by (41) g N ( z (2) )Φ( z (2) ) n + m +1 = (cid:101) C N N − (cid:89) j =1 z − a N,j − Φ( z (2) )Φ( a (2) N,j ) · Φ( z (2) ) N + m +1 . (42)Since Φ( z (2) ) = 1 /ϕ ( z ) for all z ∈ D , the last relation can be written as g N ( z (2) )Φ( z (2) ) n + m +1 = C ( N ) N − (cid:89) j =1 z − a N,j − ϕ ( z ) ϕ ( a N,j ) · ϕ n +2 ( z ) . (43)Using (43), the orthogonality relation (36) can be put in the form (cid:90) F Q n, ( x ) N − (cid:89) j =1 x − a N,j − ϕ ( x ) ϕ ( a N,j ) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) dσ ( x ) = 0 , (44)where the number N (cid:54) n is arbitrary and all points a N,j lie in D . From (44),it follows that deg Q n, = n , all zeros of the polynomial Q n, lie on (cid:98) F (whichis the convex hull of the compact set F ); besides, the gap with number ( p − c j , d j ], j = 1 , , . . . , p , may contain at most p − N (cid:54) n and arbitrary points a N,j ∈ (cid:98) F \ E , will underlie ourfurther analysis. ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 13 the GRS-method, we assume that1 n χ ( Q n, ) (cid:54)→ λ = λ F (45)as n → ∞ . We shall arrive at a contradiction by using the orthogonalityrelations (44) and condition (45).The weak compactness of the space of measures M ( (cid:98) F ) shows that1 n χ ( Q n, ) → µ (cid:54) = λ, n ∈ Λ , n → ∞ (46)for some infinite subsequence Λ ⊂ N ; besides, S ( µ ) ⊂ F , µ ∈ M ( F ), µ (1) = 1 by the above properties of the polynomial Q n, . We claim thatrelation (46) and the orthogonality relation (44) contradict each other.Setting (cid:101) V µ ( z ) := (cid:90) F log 1 | − ϕ ( z ) ϕ ( t ) | dµ ( t ) , we have P µ ( z ) = 2 V µ ( z ) − (cid:101) V µ ( z ) . Since µ (cid:54) = λ , it follows that, for z ∈ S ( µ ) ⊂ F , P µ ( z ) + ψ ( z ) (cid:54)≡ m := min z ∈ F (cid:0) P µ ( z ) + ψ ( z ) (cid:1) = P µ ( x ) + ψ ( x ) , (47)where x ∈ F . Hence there exists a point x ∈ S ( µ ), x (cid:54) = x , and a number ε > P µ ( x ) + ψ ( x ) = m > m + ε. (48)Further, since the function ψ ( z ) is harmonic and the potential P µ is lowersemi-continuous, the same inequality (48) holds in some δ -neighbourhood U δ ( x ) := ( x − δ, x + δ ) (cid:54)(cid:51) x , δ >
0, of the point x . We have x ∈ S ( µ ),and so µ ( U δ ( x )) >
0. Hence, for all sufficiently large n (cid:62) n , n ∈ Λ, thereexists a polynomial p n ( z ) = ( z − ζ n, )( z − ζ n, ) such that ζ n, , ζ n, ∈ U δ ( x )and p n divides the polynomial Q n, ; that is, Q n, /p n ∈ P n − . We set (cid:101) Q n ( z ) := Q n, ( z ) p n ( z ) = n − (cid:89) j =1 ( z − x n,j ) . (49)We may assume in what follows that, for n ∈ Λ, all zeros of the polynomial Q n, lie in the set (cid:98) F \ E . Indeed, there is at most one gap between theintervals [ c j , d j ] that may contain the interval E , in each gap lying at mostone zero of the polynomial Q n, . If some zero of the polynomial Q n, lieson the interval E , then in definition (49) of the polynomial (cid:101) Q n one shouldreplace the corresponding factor (( z − x n,j ), say) by the factor ( z − (cid:101) x n,j ),where the point (cid:101) x n,j still lies in the (open) gap, but it is not lying in E anymore. Note that, under the hypotheses of Theorem 2, the GRS-method is much easier todeal with, because an S -compact set F is a finite union of intervals of the real line and σ is a positive measure on F ; cf. [44], [18], [42]. Now in the orthogonality relation (44) we put N = n − x n,j of the polynomial (cid:101) Q n as points a N,j (with the possible correctionmentioned above), relation (44) assuming the form0 = (cid:90) F \ U δ ( x ) Q n, ( x ) p n ( x ) n − (cid:89) j =1 − ϕ ( x ) ϕ ( x n,j ) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) dσ ( x )+ (cid:90) U δ ( x ) Q n, ( x ) p n ( x ) n − (cid:89) j =1 − ϕ ( x ) ϕ ( x n,j ) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) dσ ( x ) . (50)We denote by I n, and I n, , respectively, the first and second integrals in (50).Since the integrand in I n, has constant sign for x ∈ F \ U δ ( x ), we have | I n, | = (cid:90) F \ U δ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) Q n, ( x ) p n ( x ) n − (cid:89) j =1 − ϕ ( x ) ϕ ( x n,j ) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dσ ( x )= (cid:90) F \ U δ ( x ) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j )1 − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) dσ ( x ) . (51)A similar analysis (see Lemma 7 of [18]) with the use of standard machineryof the logarithmic potential theory shows thatlim n →∞ n ∈ Λ | I n, | /n = exp (cid:26) − min x ∈ F \ U δ ( x ) (cid:0) P µ ( x ) + ψ ( x ) (cid:1)(cid:27) = e − m . (52)We give a proof of (52) for completeness (cf. Lemma 7 of [18]).Indeed, − n n − (cid:88) j =1 log | − ϕ ( x ) ϕ ( x n,j ) | → (cid:90) F log 1 | − ϕ ( x ) ϕ ( t ) | dµ ( t ) = (cid:101) V µ ( x ) (53)as n → ∞ uniformly in x ∈ F . Hence,min x ∈ F (cid:26) − n log (cid:18) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) (cid:19)(cid:27) → min x ∈ F (cid:8) P µ ( x ) + ψ ( x ) (cid:9) (54)as n → ∞ . As a result, we havemax x ∈ F (cid:26) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) (cid:27) /n → exp (cid:8) − min x ∈ F (cid:2) P µ ( x ) + ψ ( x ) (cid:3)(cid:9) (55)as n → ∞ , proving thereby the upper estimatelim n →∞ n ∈ Λ | I n, | /n (cid:54) e − m . ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 15
Let us now prove the corresponding lower estimate. The potential P µ isweakly continuous, and hence the function P µ + ψ is approximately contin-uous with respect to the Lebesgue measure on the compact set F . Conse-quently, for any ε >
0, the set e = { x ∈ F : ( P µ + ψ )( x ) < m + ε } has positive Lebesgue measure. From our assumptions we have − n log (cid:26) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) (cid:27) → ( P µ + ψ )( x )as n → ∞ with respect to the measure on F . So, the measure of the set e n := (cid:26) x ∈ e : − n log (cid:18) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) (cid:19) < m + ε (cid:27) tends to the measure of e as n → ∞ . Hencelim n →∞ n ∈ Λ | I n, | /n (cid:62) e − ( m + ε ) lim n →∞ n ∈ Λ (cid:18)(cid:90) e n ϕ (cid:48) ( x ) dσ ( x ) (cid:19) /n = e − ( m + ε ) , (56)the last equality in (56) holding because σ (cid:48) ( x ) > F . The lowerestimate lim n →∞ n ∈ Λ | I n, | /n (cid:62) e − m follows from (56), because ε > I n, we have the estimate | I n, | (cid:54) (cid:90) U δ ( x ) | Q n, ( x ) | n − (cid:89) j =1 (cid:12)(cid:12)(cid:12)(cid:12) x − x n,j − ϕ ( x ) ϕ ( x n,j ) (cid:12)(cid:12)(cid:12)(cid:12) · ϕ (cid:48) ( x ) ϕ n +2 ( x ) dσ ( x ) . (57)Now an analysis similar to that above shows thatlim n →∞ n ∈ Λ | I n, | /n (cid:54) exp (cid:26) − min x ∈ U δ ( x ) (cid:0) P µ ( x ) + ψ ( x ) (cid:1)(cid:27) (cid:54) e − m < e − ( m + ε ) . (58)But relations (52) and (58) contradict the equality I n, = − I n, , which isconsequent on the orthogonality relations (44).This proves Theorem 2. Remark 1.
In a certain sense, the above transformations mean the changeof the variable z by the variable ζ = ϕ ( z ). For the case of the Riemannsurface w = z − f ( z ) := (cid:90) − r ( x )( z − x ) dx √ − x , where r ∈ C ( z ) is an arbitrary complex rational function without poles andzeros on E , such a simplification does not apply anymore, but the conclusions of Theorem 2 remain valid . It is also worth pointing out the role of theRiemann surface of the function w = z − w = z − g = 0, use is made of theRiemann surface of the function w = ( z − e ) . . . ( z − e ) of genus g = 1.A generalization of the results obtained here to the elliptic case (of course,if such an extension will come to being) will be of the utmost importancein assessing the potency of the method proposed here when investigatingthe general case of a pair of functions f , f ∈ A ◦ (Σ). Of course, in thisgeneral case the problem of the formula for strong asymptotics for Pad´epolynomials valid for an arbitrary function f from the class A ◦ (Σ) will havea great value; see [37], [31], [5] in this respect. Remark 2.
It is well known (see [1], and also [4] and [29]) that, for a pairof functions f , f forming a Nikishin system, the support of the equilibriummeasure λ in the diagonal case (which is considered here) coincides with theentire compact set F ; this means that only the case of identical equality inrelations (16) is possible. So far, this fact has not yet been proved withinthe framework of the approach proposed here. Remark 3.
It is worth pointing out that the approach proposed in thepresent paper stems, to some extent, from the analysis of numerical experi-ments of [21]–[23]. As was already mentioned above, the author intends to investigate this general casein a separate paper; see [50].
ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 17 -2-1.5-1-0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
Figure 1.
Zeros of diagonal Hermite–Pad´e polynomials of type I Q , (blue points), Q , (red points), Q , (black points) for the tuple of func-tions [1 , f , f ], where f ( z ) := ( z − − / , f ( z ) := (cid:0) ( z − . − . i )( z + . − . i ) (cid:1) − / , forming an Angelesco system. No theoretical justification of suchbehavior of the zeros of Hermite–Pad´e polynomials of type I has not yetbeen found to date. -2-1.5-1-0.5 0 0.5 1 1.5 2-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 Figure 2.
Zeros of the denominator of diagonal Hermite–Pad´e approxi-mants of type II P (light blue points) for the tuple of functions [1 , f , f ],where f ( z ) := ( z − − / , f ( z ) := (cid:0) ( z − . − . i )( z + . − . i ) (cid:1) − / , formingan Angelesco system. Theoretical justification of such behavior of zeros ofHermite–Pad´e polynomials of type II was obtained in [2]. ISTRIBUTION OF THE ZEROS OF HERMITE–PAD´E POLYNOMIALS 19
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