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Christoffel formula for kernel polynomials on the unit circle C.F. Braccialia, A. Mart´ınez-Finkelshteinb, A. Sri Rangaa∗and D.O. Veronesec a DMAp, IBILCE, UNESP - Universidade Estadual Paulista, 7 15054-000,Saõ José do Rio Preto, SP, Brazil. 1 0 a Departamento de Matema´ticas, Universidad de Almer´ıa, 04120 Almer´ıa, 2 and Instituto Carlos I de F´ısica Teo´rica and Computacional, Granada University, Spain n a cICTE, UFTM - Universidade Federal do Triaˆngulo Mineiro, J 38064-200Uberaba, MG, Brazil. 8 1 January 19, 2017 ] A C . h Abstract t a m Given a nontrivial positive measure µ on the unit circle, the associated n [ Christoffel-Darboux kernels are Kn(z,w;µ) = k=0ϕk(w;µ)ϕk(z;µ), n ≥ 0, where ϕk(·;µ) are the orthonormalpolynomials with respect to the measure µ. 1 P Letthepositivemeasureν ontheunitcirclebegivenbydν(z)=|G2m(z)|dµ(z), v 5 where G2m is a conjugate reciprocalpolynomialofexact degree2m. We estab- 9 lish a determinantal formula expressing {Kn(z,w;ν)}n≥0 directly in terms of 9 {Kn(z,w;µ)}n≥0. 4 Furthermore, we consider the special case of w =1; it is knownthat appro- 0 priately normalized polynomials Kn(z,1;µ) satisfy a recurrence relation whose . 1 coefficients are given in terms of two sets of real parameters {cn(µ)}∞n=1 and 0 {gn(µ)}∞n=1,with0<gn <1forn≥1. Thedoublesequence{(cn(µ),gn(µ))}∞n=1 7 characterizesthemeasureµ. Anaturalquestionabouttherelationbetweenthe 1 : parameterscn(µ),gn(µ),associatedwithµ,andthesequencescn(ν),gn(ν),cor- v responding to ν, is also addressed. i X Finally, examples are considered, such as the Geronimus weight (a measure supported on an arc of T), a class of measures given by basic hypergeometric r a functions,andaclassofmeasureswithhypergeometricorthogonalpolynomials. Keywords: Orthogonal functions, Christoffel formulas, three term recurrence relation, orthogonal polynomials on the unit circle. 1 Introduction Given a nontrivial positive measure µ on the unit circle T := {ζ = eiθ: 0 ≤ θ ≤ 2π} the associated orthonormal polynomials ϕ (z;µ) = κ zn +lower degree terms are n n defined by κ = κ (µ)> 0 and n n 2π ϕ (ζ;µ)ϕ (ζ;µ)dµ(ζ) = ϕ (eiθ;µ)ϕ (eiθ;µ)dµ(eiθ)= δ , m n m n m,n ZT Z0 ∗[email protected] (corresponding author) for m,n = 0,1,2,..., where δ stands for the Kronecker delta. These are m,n orthogonal polynomials on the unit circle, or in short, OPUC. A recent complete treatise on OPUC is the monograph [24]. Among their fundamental properties is that all their zeros belong to the open unit disk D := {z ∈ C: |z| < 1}. The reproducing kernels K (z, w;µ) (also known as Christoffel–Darboux kernels n or simply CD kernels) associated with the measure µ are given by n K (z,w;µ) = ϕ (w;µ)ϕ (z;µ), n≥ 0. (1.1) n j j j=0 X They have been the subject of study in many recent contributions including the review[25]ontheiruseinthespectraltheory oforthogonalpolynomialsandrandom matrices. In what follows we use the standard notation for the reversed (or conjugate- reciprocal) polynomials: if q is an algebraic polynomial of degree n, then q∗(z) := znq(1/z). Withthisnotation, thewell-known Christoffel-Darbouxformulasays thatforz 6= w, ϕ (w;µ)ϕ (z;µ)−ϕ∗ (w;µ)ϕ∗ (z;µ) K (z,w;µ) = n+1 n+1 n+1 n+1 n wz−1 (1.2) wzϕ (w;µ)ϕ (z;µ)−ϕ∗(w;µ)ϕ∗(z;µ) = n n n n , n≥ 0. wz−1 Notice that K (z,0;µ) = ϕ∗(0;µ)ϕ∗(z;µ). On the other hand, if w ∈ T, then all n n n zeros ofK (z,w;µ) (asapolynomialinz)lieonT, anduptoanormalization factor, n (wz −1)K (z,w;µ) is a so-called para-orthogonal polynomial of degree n+1. For n information concerning para-orthogonal polynomials we refer to [4] and references therein. Amultiplicationofthegivenmeasureµbyafactorthatispositiveonitssupport, supp(µ), yields a new measure and a correspondingset of OPUC and of CD kernels. It is a natural question to ask whether there is an explicit connection between these two sets. In this paper we are interested in the case when the factor is of the form |g|2, whereg is a polynomial. For the orthogonality on thereal axis, this is thecontent of the so-called Christoffel formula (see, for example, [29]), which was extended in [20] to cover OPUC (see also [22], which generalizes [20] and constitutes a nice survey of related results obtained prior to 1999, as well as some recent related results in [1, 2]). In these cases there is a determinantal expression for the “new” orthogonal polynomials in terms of those orthogonal with respect to µ. One of the goals of this paper is to obtain such a determinantal formula for the CD kernels on T. Observe that this kind of expressions is not a trivial consequence of the analogous formulas for OPUC. RecallthattheclassicalFejér–Riesztheorem(see[19,§1.12])saysthateverynon- negative trigonometric polynomial f(θ) can be written as |g(z)|2, z = eiθ, where g is an algebraic polynomial non-vanishing in D. Equivalently, we can say that f(θ) is of the form z−mG(z), where G is a self-reciprocal polynomial (i.e., G∗ = G) of degree 2m. Motivated by this result, we slightly weaken the assumptions of Fejér and Riesz and require the trigonometric multiplication factor of µ to be non-negative only 2 within the support of µ. More precisely, let G be a self-reciprocal polynomial of 2m exact degree 2m, m ∈ N, and non-negative on supp(µ), and let G (ζ) dν(ζ)= 2m dµ(ζ)= |G (ζ)|dµ(ζ), ζ ∈ T, (1.3) ζm 2m which is also a positive measure on T. We denote by z ,z ,...,z 1 2 2m the zeros of G . Those of them not on T must appear in pairs symmetric with 2m respect to T; notice that no z is = 0. However, unlike in the case of Fejér–Riesz, j if supp(µ) 6= T, zeros of G on T also can be simple, as long as the hypothe- 2m sis of positivity of G (ζ)/ζm on supp(µ) (i.e., the positivity of ν on supp(µ)) is 2m preserved.1 Inwhatfollows wewillbemainly interested inthecase whenall z ’sarepairwise j distinct (or equivalently, when all zeros of G are simple). 2m Definition 1.1. Given m ∈ N, we call a set P = {p ,p ,...,p } of 2m+1 not 0 1 2m identically 0 algebraic polynomials p admissible if p (z) ≡ 1, p (z) = zm, j 0 2m min{j,m} p (z) = b zi, j = 1,...,2m−1, (1.4) j ij j=max{0,j−m} X and either one of the following three conditions is satisfied: degp < j for j = 1,...,2m, (1.5) j degp < m for j = 1,...,2m−1, (1.6) j or p (0) = 0 for j = 1,...,2m. (1.7) j Theorem 1.1. For an admissible set P = {p ,p ,...,p } and w ∈ C define 0 1 2m Q (z,w) := p (z)K (z,w;µ), j = 0,1,...,2m, (1.8) j j n+2m−j and the (2m+1)×(2m+1) matrix Q (z,w) Q (z,w) ... Q (z,w) 0 1 2m Q (z ,w) Q (z ,w) ... Q (z ,w) 0 1 1 1 2m 1 Q(z,w) = Q(P)(z,w) :=  .. .. .. .. . (1.9) . . . .   Q (z ,w) Q (z ,w) ... Q (z ,w) 0 2m 1 2m 2m 2m     (P) Then, there exists a polynomial C (w) = C (w) of degree ≤ (2m+1)(n+m) such n n that detQ(z,w) = C (w)G (z)K (z,w;ν). (1.10) n 2m n 1For instance, if supp(µ) = {eiθ : 0 < θ1 ≤ θ ≤ θ2 < 2π}, we can consider the self reciprocal polynomial G2(z) = e−iα/2e−iβ/2(z−eiα)(z−eiβ), with 0 < α ≤ θ1 and θ2 ≤ β < 2π. Then the rational function G2(ζ) G2(eiθ) θ−α β−θ = =4sin sin ζ eiθ 2 2 is positive on supp(µ), butnot on theentire T;see Example 4.3 in Section 4 below. 3 Observe that for certain values of w, both sides of (1.10) can vanish, in which casetheidentity in(1.10)isformallycorrect, butpractically useless. Thus,anatural question is about sufficient conditions for C 6= 0. n Theorem 1.2. Let all the zeros of G be simple. For an admissible set P = 2m {p ,p ,...,p } and w ∈ C, with the notations of Theorem 1.1, if either 0 1 2m i) |w| ≥ 1, condition (1.5) holds and polynomials Q (·,w),Q (·,w),...,Q (·,w) 1 2 2m are linearly independent, or ii) |w| ≤ 1, condition (1.6)holdsandpolynomials Q (·,w),Q (·,w),...,Q (·,w) 0 1 2m−1 are linearly independent, or iii) 0< |w| ≤ 1, condition (1.7) holds and polynomials Q (·,1/w), Q (·,1/w), ..., 1 2 Q (·,1/w) are linearly independent, 2m (P) then C (w)6= 0. n Remark 1.1. If the polynomial G has non-simple zeros, then the results above 2m stillholdifonereplacesthepolynomialsineachrowofthematrixQbytherespective derivativesinaccordancewiththeorderofmultiplicity. Forexample,ifz 6= z = z , 1 2 3 then the fourth row of Q must be replaced by Q′(z ,w),Q′(z ,w),··· ,Q′ (z ,w),Q′ (z ,w), 0 2 1 2 2m−1 2 2m 2 where Q′ stands for its derivative with respect to z. j Given an admissible set P = {p ,p ,...,p }, we define P = {p ,p ,...,p }, 0 1 2m 0 1 2m with pj(z) := zjpj(1/z) =zj−deg(pj)p∗j(z), j = 0,1,.b..,2mb. b (b1.11) Observe that (P)= P. b Propositiond1b.3. A set of polynomials P = {p0,p1,...,p2m} is admissible if and only if P = {p ,p ,...,p } is. Moreover, P satisfies (1.5) (resp., (1.7)), then P 0 1 2m satisfies (1.7) (resp., (1.5)). Addibtionalbly, bif w 6= 0b, b b 1 C(P)(w) 6= 0 ⇔ C(P) 6= 0. (1.12) n n w (cid:18) (cid:19) It would be nice to have a simple recipe for constructing an admissible set P for which (1.10) renders a non-trivial identity for the CD kernel K (z,w;ν). Obviously, n there is no “universal” P such that = C(P)(w) in (1.10) is 6= 0 for all w ∈ C. n However, there is a simple admissible set that guarantees this, at least for |w| = 1. It is easy to check that P = {p ,p ,...,p }, with 0 1 2m zj/2, if j is even, p (z) = z⌊j/2⌋ = j = 0,1,...,2m, (1.13) j (z(j−1)/2, if j is odd, isadmissibleandsatisfiesboth(1.5)and(1.6). ThecorrespondingP = {p ,p ,...,p } 0 1 2m is p (z) = z⌊j+1/2⌋ = zj/2, if j is even, j = 0,1,...,b2m, b b (1.14b) j (z(j+1)/2, if j is odd, satisfyinbg, by Proposition 1.3, condition (1.7). 4 Proposition 1.4. Let the admissible set of polynomials P = {p ,p ,...,p } be 0 1 2m given by (1.13), and let w be such that K (0,w;µ) 6= 0 and degK (·,w;µ) = n for n ∈ N. (1.15) n n Then Q (·,w), Q (·,w), ..., Q (·,w) are linearly independent. In particular, it 0 1 2m holds for |w| = 1. Corollary 1.5. For the admissible sets of polynomials P and P, given by (1.13) b (P) (P) and (1.14), respectively, both C (w) 6= 0 and C (w) 6= 0 in (1.10) when |w| = 1. n n b Example 1.1. Let us consider the normalized Lebesgue measure on T, 1 dµ(ζ)= |dζ|, ζ ∈ T. (1.16) 2π Then 1−wn+1zn+1 K (z,w;µ) = , n ≥ 0, n 1−wz so that all w 6= 0 satisfy conditions (1.15) from Proposition 1.4. In particular, for all (P) such w, and for the admissible set P given by (1.13), C (w) 6= 0 in (1.10). How- n ever, K (z,0;µ) ≡ 1, which implies that polynomials Q (z,0) in (1.8) are linearly n j dependent for any choice of the admissible set of polynomials P, and hence, one cannot find an admissible set for which detQ(·,0) 6≡ 0. Clearly, we still can recover K (z,0;ν) = ϕ∗(0;ν)ϕ∗(z;ν) by taking limit, n n n 1 G (z)K (z,0;ν) = lim detQ(z,w). 2m n w→0C(P)(w) n The proofs of the assertions above are gathered in Section 2. In Section 3 we consider an interesting particular case of w = 1, for which K (·,1;µ) constitute an instance of paraorthogonal polynomials on T. A convenient n “symmetrization” of these polynomials was found in [8]; it was shown there that the appropriately normalized K (·,1;µ), that we denote by R (·;µ) (see the precise n n definition in Section 3) satisfy a three term recurrence relation of the form R (z;µ) = [(1+ic )z+(1−ic )]R (z;µ) n+1 n+1 n+1 n (1.17) − 4(1−g )g zR (z;µ), n n+1 n−1 for n ≥ 0, with R (z) = 0 and R (z) = 1. Sequences {c }∞ = {c (µ)}∞ −1 0 n n=1 n n=1 and {g }∞ = {g (µ)}∞ are both real, with 0 < g < 1 for n ≥ 1. As shown in n n=1 n n=1 n [4,6,8],thedoublesequence{(c (µ),g (µ))}∞ isaparametrization ofthemeasure n n n=1 µ, alternative to its Verblunsky coefficients. Thus, a natural question is the relation between the parameters c (µ), g (µ), associated with µ, and the sequences c (ν), n n n g (ν), correspondingto ν. These questions will be addressed in Section 3. Since the n statement of the corresponding results requires introducing a considerable piece of notation, we postpone it to the aforementioned section. Finally, in Section 4 we consider four different applications of our formulas: a rather straightforward case when µ is the Lebesgue measure on T, the Geronimus weight (a measure supported on an arc of T), a class of measures given by basic hypergeometric functions, and a class of measures with hypergeometric OPUC. 5 2 Proof of the Christoffel formula for kernels First we discuss a characterization of the kernel polynomials K . n Let w ∈ C be fixed. With the positive measure µ on T we consider the complex- valued measure on T given by, dµ (ζ) =(1−wζ)dµ(ζ). w The following simple lemma will be useful in the forthcoming proofs: Lemma 2.1. Let f be an integrable function on T such that either one of the following condition is satisfied: i) |w| ≤ 1 and |f(ζ)|dµ (ζ) = 0, (2.1) w Z or ii) |w| ≥ 1 and ζ|f(ζ)|dµ (ζ) = 0. (2.2) w Z Then f = 0 µ-a.e. (in case when |w| =6 1) and µ -a.e., otherwise. T\{w} In particular, if f is a polynomial and µ has an infinite number of points of (cid:12) increase, then i) or ii) imply that f ≡ 0. (cid:12) Proof. Consider i) first. For w = 0 the statement is trivial, so let 0 < |w| ≤ 1. By assumptions of the lemma, 1 |f(ζ)| −ζ dµ(ζ)= 0, w Z (cid:18) (cid:19) and hence, 1 |f(ζ)| −eiθζ dµ(ζ) = 0, θ = −arg(w). |w| Z (cid:18) (cid:19) In particular, taking the real part, we get 1 |f(ζ)|Re −eiθζ dµ(ζ) = 0, |w| Z (cid:18) (cid:19) and it remains to notice that 1 Re −eiθζ > 0, |w| (cid:18) (cid:19) unless |w| = 1 and ζ =w. In the case ii), we have that |f(ζ)| w−ζ dµ(ζ) = |f(ζ)|(w−ζ)dµ(ζ) =0, Z Z (cid:0) (cid:1) so that |f(ζ)| |w|−eiθζ dµ(ζ) = 0, Z (cid:16) (cid:17) and again, Re |w|−eiθζ > 0, (cid:16) (cid:17) unless |w| = 1 and ζ =w. 6 Lemma 2.2. For a fixed w ∈ C and n ∈ N, the CD kernel K (z,w;µ) is a poly- n nomial in z of degree ≤ n, characterized up to a constant factor by the following orthogonality relations, ζsK (ζ,w;µ)dµ (ζ)= 0, 1 ≤ s ≤ n, (2.3) n w Z and the additional condition a) if |w| ≥ 1, then K (z,w;µ) is of degree exactly n; n b) if |w| ≤ 1, then K (ζ,w;µ)dµ (ζ)6= 0. n w Z Proof. Orthogonalityconditions(2.3)areastraightforwardconsequenceof (1.2)and of the well-known relations for the reversed polynomials, ζsϕ∗(ζ;µ)dµ(ζ) = 0, 1 ≤ s ≤ n, and ϕ∗(ζ;µ)dµ(ζ) 6= 0. n n Z Z Since for |w| ≥ 1, ϕ (w;µ) 6= 0, usingthe definition (1.1) of K (z,w;µ) we conclude n n that it is a polynomial in z of degree = n. For |w| ≤ 1, using (1.2) and the fact that ϕ∗ does not vanish inside or on the unit disk, we see that n+1 K (ζ,w;µ)dµ (ζ)= K (ζ,w;µ)(1−wζ)dµ(ζ) n w n Z Z = ϕ∗ (w;µ) ϕ∗ (ζ;µ)dµ(ζ) 6= 0. n+1 n+1 Z Let us prove that these relations characterize the CD kernel. Assume that |w| ≥ 1. If P is a polynomial of degree exactly n, satisfying ζsP(ζ)dµ (ζ)= 0, 1 ≤ s≤ n, (2.4) w Z then there exists a constant c6= 0 such that L(z) := cK (z,w;µ)−P(z) is of degree n ≤ n−1. By hypothesis, L satisfies the same orthogonality conditions, so that 0 = ζL(ζ)L(ζ)dµ (ζ)= ζ|L(ζ)|2µ (ζ), w w Z Z and it remains to apply Lemma 2.1, ii), to conclude that L ≡ 0. On the other hand, if P of degree ≤ n satisfies (2.4) and is such that P(ζ)dµ (ζ)6= 0, w Z then there exists a non-zero constant, let us denote it by c again, such that (P(ζ)−cK (ζ,w;µ))dµ (ζ)= 0. n w Z Combining it with (2.3) and (2.4) we get that for s = 0,1,...,n, ζs(P(ζ)−cK (ζ,w;µ))dµ (ζ)= ζs(P(ζ)−cK (ζ,w;µ))(1−wζ)dµ(ζ)= 0. n w n Z Z It means that (P(ζ)−cK (ζ,w;µ))(1−wζ)= constϕ (ζ;µ). n n+1 Since ϕ cannot vanish at 1/w, we conclude that P(ζ)≡ cK (ζ,w;µ). n+1 n 7 In order to prove Theorem 1.1 we need some preparatory steps. With the notation of Section 1, it is immediate to check that detQ(z,w) is an algebraic polynomialinz (of degree≤ (n+2m)) andinw (ofdegree≤ (m+n)(2m+ 1)). Furthermore, for each w ∈ C it vanishes at the zeros z ,...,z of G . Thus, 1 2m 2m we can write detQ(z,w) = G (z)A (z,w), (2.5) 2m n where A is an algebraic polynomial in z (of degree ≤ n) and in w. We need to n show that for each w ∈ C, there exists a constant C such that A (z,w) =CK (z,w;ν). (2.6) n n If this is established, the polynomial dependence of C from w (as well as on the admissible set P chosen, see Definition 1.1) is a straightforward consequence of (2.5)–(2.6). We prove (2.6) by appealing to the characterization of K given in Lemma 2.2. n By (2.3) and the definition of ν, kernels K (z,w;ν) satisfy n ζsG (ζ)K (ζ,w;ν)dµ (ζ) = 0, m+1 ≤ s ≤ m+n. 2m n w Z Thus, a necessary condition for (2.6) is that ζsG (ζ)A (ζ)dµ (ζ) = 0, m+1 ≤ s ≤ m+n. 2m n w Z This is always true, and it is an immediate consequence of the following lemma: Lemma 2.3. For j = 0,1,...,2m, ζsQ (ζ,w)dµ (ζ) = 0, s = m+1,...,m+n. (2.7) j w Z Thus, ζsdetQ(ζ,w)dµ (ζ) =0, m+1 ≤ s ≤ m+n. (2.8) w Z Proof. In order to calculate ζsp (ζ)K (ζ,w;µ)dµ (ζ) for m+1 ≤ s ≤ m+n, 0≤ j ≤ 2m, (2.9) j n+2m−j w Z with account of (1.4), it is sufficient to find the values of ζs−rK (ζ,w;µ)dµ (ζ) for m+1 ≤ s ≤ m+n, 0≤ j ≤ 2m, (2.10) n+2m−j w Z for max{0,j −m}≤ r ≤ min{j,m}. Since r ≤ min{j,m} ≤ m, we get s−r ≥ m+1−r ≥ 1. Analogously, from r ≥ max{0,j −m}≥ j −m, we conclude that s−r ≤ m+n−r ≤ n+2m−j. By (2.3) it follows that all integrals in (2.10) (and consequently, in (2.9)) vanish, which yields (2.7)–(2.8). 8 So, the necessary condition (orthogonality) always holds. Now we go for a sufficient condition, given by a) and b) of Lemma 2.2. Recall first the following well known fact, that we state just as a remark. Remark 2.1. By the maximum principle, |ϕ∗(z;µ)| > |ϕ (z;µ)| for |z| < 1, and n n |ϕ∗(z;µ)| < |ϕ (z;µ)| for |z| > 1. By (1.2), n n K (z,w;µ) = 0 ⇔ ϕ (w;µ)ϕ (z;µ)−ϕ∗ (w;µ)ϕ∗ (z;µ) = 0, n n+1 n+1 n+1 n+1 which for |w| > 1 can be rewritten as ϕ (z;µ) ϕ∗ (w;µ) n+1 = n+1 . ϕ∗ (z;µ) ϕ (w;µ) n+1 (cid:18) n+1 (cid:19) The right hand side is of absolute value < 1, so that equality is possible only for |z| < 1. Same analysis is valid for the other case and we conclude that the zeros of K (z,w;µ) (as a polynomial of z) are of absolute value > 1 (respectively, = 1 or n < 1) if |w| < 1 (respectively, |w| = 1 or |w| < 1). Lemma 2.4. Let A be defined by (2.5) and w ∈ C fixed. The following conditions n are necessary and sufficient for A (·,w) ≡ 0: n i) |w| ≥ 1 and deg(detQ(·,w)) < n+2m; (2.11) ii) |w| ≤ 1 and ζmdetQ(ζ,w)dµ (ζ) = 0. (2.12) w Z Proof. A (·,w) ≡ 0is clearly asufficientcondition in i) andii) for(2.11)and(2.12), n respectively. So, we prove that this is also necessary. For i), if degA (·,w) < n (which is equivalent to deg(detQ(·,w)) < n+2m), n then by (2.8), 0 = ζm+1A (ζ,w)detQ(ζ,w)dµ (ζ)= ζ|A (ζ,w)|2|G (ζ)|dµ (ζ), n w n 2m w Z Z and we use again Lemma 2.1, ii), to conclude that A (·,w) ≡ 0. n For ii), if ζmdetQ(ζ,w)dµ (ζ) = A (ζ,w)|G (ζ)|dµ (ζ)= 0, w n 2m w Z Z then by (2.8) we have in fact that for s = 0,1,...,n, ζsA (ζ,w)|G (ζ)|dµ (ζ)= ζsA (ζ,w)(1−wζ)|G (ζ)|dµ(ζ) = 0. n 2m w n 2m Z Z Since degA (·,w) ≤ n, we conclude that n A (ζ,w)(1−wζ)= cϕ (ζ;ν), n n+1 But ϕ (ζ;ν) cannot vanish at ζ = 1/w ∈/ D, which implies that c = 0, and n+1 A (·,w) ≡ 0. n 9 A combination of Lemmas 2.3 and 2.4 constitutes the proof of Theorem 1.1. Indeed,let|w| ≥1. Considertheidentity(2.5);byLemma2.4,i),eitherA (·,w) ≡ 0 n or degA (·,w) = n. In the latter case by the characterization in Lemma 2.2, a), n A (z,w) = CK (ζ,w;ν). n n On the other hand, if |w| ≤ 1, then again by Lemma 2.4, ii), either A ≡ 0 or n A (ζ)|G (ζ)|dµ (ζ)6= 0, n 2m w Z in which case by Lemma 2.2, b), A coincides, up to a constant factor, with n K (·,w;ν). n Now we turn to Theorem 1.2. Checking(2.11)or(2.12)isnotstraightforward. Seekingamoreexplicitalgebraic condition, weintroduceanotation fortheminorsofthematrixQ: theone, obtained by deleting its first row and column, Q (z ,w) ... Q (z ,w) 1 1 2m 1 ∆0(w) = det ... ... ... , (2.13) Q (z ,w) ... Q (z ,w) 1 2m 2m 2m     and the minor obtained from Q by deleting its first row and its last column, Q (z ,w) ... Q (z ,w) 0 1 2m−1 1 ∆m(w) = det ... ... ... . (2.14) Q (z ,w) ... Q (z ,w) 0 2m 2m−1 2m     Lemma 2.5. Let A be defined by (2.5) and w ∈ C fixed, and let either one of the n following conditions hold: • (1.5) with |w| ≥ 1; • (1.7) with |w| ≤ 1. Then A (·,w) ≡ 0 ⇔ ∆ (w) = 0. n 0 On the other hand, if condition (1.6) holds with |w| ≤1, then A (·,w) ≡ 0 ⇔ ∆ (w) = 0. n m Proof. Under assumptions of the first part, observe that G (z)A (z,w) = detQ(z,w) = ∆(n)(w)K (z,w)+span {Q ,...,Q }. 2m n m n+2m 1 2m (2.15) We prove the (⇒) part first, assuming ∆ (w) 6= 0. If (1.5), then the lead- 0 ing coefficient of detQ(·,w) is ∆(n)(w)× leading coefficient of K (·,w) 6= 0, so m n+2m A (·,w) 6≡ 0. And if (1.7) takes place, then (see Remark 2.1) n detQ(0,w) = ∆ (w)K (0,w) 6= 0, 0 n+2m so A (·,w) 6≡ 0 again. n 10