Asymptotics of matrix valued orthogonal polynomials on 11 Alfredo Dea no1 Arno B.J. Kuijlaars2 and Pablo Rom an3

2025-05-06 0 0 825.35KB 48 页 10玖币

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Asymptotics of matrix valued orthogonal polynomials on

[−1,1]

Alfredo Dea˜no∗1, Arno B.J. Kuijlaars†2, and Pablo Rom´an‡3

1Department of Mathematics, Universidad Carlos III de Madrid, Spain

2Department of Mathematics, Katholieke Universiteit Leuven, Leuven, Belgium

3FaMAF-CIEM, Universidad Nacional de C´ordoba, Argentina

April 11, 2023

Abstract

We analyze the large degree asymptotic behavior of matrix valued orthogonal poly-

nomials (MVOPs), with a weight that consists of a Jacobi scalar factor and a matrix

part. Using the Riemann–Hilbert formulation for MVOPs and the Deift–Zhou method

of steepest descent, we obtain asymptotic expansions for the MVOPs as the degree

tends to inﬁnity, in diﬀerent regions of the complex plane (outside the interval of

orthogonality, on the interval away from the endpoints and in neighborhoods of the

endpoints), as well as for the matrix coeﬃcients in the three-term recurrence relation

for these MVOPs. The asymptotic analysis follows the work of Kuijlaars, McLaughlin,

Van Assche and Vanlessen on scalar Jacobi-type orthogonal polynomials, but it also

requires several diﬀerent factorizations of the matrix part of the weight, in terms of

eigenvalues/eigenvectors and using a matrix Szeg˝o function. We illustrate the results

with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group

theory.

1 Introduction and statement of results

1.1 Introduction

In this paper, we are interested in the large degree asymptotic behavior of matrix

valued orthogonal polynomials (MVOPs), with orthogonality deﬁned on [−1,1]. The

weight matrix Won [−1,1] is of size r×r, and we take it of the form

W(x) = (1 −x)α(1 + x)βH(x) (1.1)

with α, β > −1 and where the matrix valued function H(x) satisﬁes the following:

Assumption 1.1. (a) H(x) is an r×rcomplex valued matrix for x∈[−1,1],

(b) H(x) is Hermitian positive deﬁnite for x∈(−1,1),

(d) H(−1) and H(1) are not identically zero.

∗alfredo.deanho@uc3m.es

†arno.kuijlaars@kuleuven.be

‡pablo.roman@unc.edu.ar

arXiv:2210.00797v2 [math.CA] 10 Apr 2023

The real analyticity means that Hhas an analytic extension to a neighborhood of

[−1,1] in the complex plane that we will also denote by H. The requirement in (b)

is that H(x) is a Hermitian matrix, i.e., H(x) = H(x)∗, for every x∈(−1,1), with

positive eigenvalues. Then by real analyticity H(−1) and H(1) are Hermitian non-

negative deﬁnite, but not necessarily positive deﬁnite, as some of the eigenvalues could

vanish at ±1. However, not all eigenvalues can vanish because of the requirement in

(d). In our examples the matrix valued function His polynomial in x, and H(±1)

will be singular.

If H(x) is a diagonal matrix for every x∈[−1,1], then the MVOPs reduce to rusual

scalar orthogonal polynomials with weight functions of the type w(x) = (1 −x)α(1 +

x)βh(x), where h(x) is analytic in a neighborhood of [−1,1]. Strong asymptotics for

these kind of orthogonal polynomials was obtained with Riemann-Hilbert methods by

Kuijlaars, McLaughlin, Van Assche and Vanlessen in [32], and the present paper can

be viewed as a matrix valued extension of that work.

The monic MVOP Pnis deﬁned by the property that for m, n ≥0,

−1

Pn(x)W(x)Pm(x)∗dx =δn,mΓn(1.2)

with a positive deﬁnite matrix Γn, where Pn(x) = xnIr+··· is a matrix valued

polynomial of degree nwhose leading coeﬃcient is the identity matrix Ir. The integral

in (1.2) is taken entrywise. Under Assumption 1.1, existence and uniqueness of the

sequence (Pn)nis guaranteed.

Matrix orthogonal polynomials have appeared in many diﬀerent contexts in the

literature in the last years. Following classical ideas in the scalar case, Dur´an and

Gr¨unbaum in [18, 19] studied MVOPs from the perspective of eigenfunctions of second

order diﬀerential operators with matrix coeﬃcients. This work has produced a large

number of contributions in the literature, extending classical identities for scalar OPs

to the matrix case. A general analysis of the matrix Bochner problem (the classiﬁcation

of N×Nweight matrices whose associated MVOPs are eigenfunctions of a second

order diﬀerential operator) has been recently addressed by Casper and Yakimov in [9],

using techniques from noncommutative algebra.

From the point of view of group theory and representation theory, the study of

matrix valued spherical functions has led to families of MVOPs associated to com-

pact symmetric spaces. The ﬁrst example of this connection is given by Gr¨unbaum,

Pacharoni and Tirao in [24] for the symmetric pair (G, K) = (SU(3),U(2)), see also

[34, 35, 37]. Another approach was developed in [28, 29] for the (SU(2) ×SU(2),diag),

and later extended to a more general set-up in the context of the so-called multiplicity

free pairs. In particular, [27] gives a detailed study of the Gegenbauer matrix val-

ued orthogonal polynomials, which can be considered as matrix valued analogues of

the Chebyshev polynomials, i.e., the spherical polynomials on (SU(2) ×SU(2),diag),

better known as the characters on SU(2), see also [1] for the quantum group case.

The Riemann–Hilbert formulation for MVOPs appears in the works of Gr¨unbaum,

de la Iglesia and Mart´ınez-Finkelshtein [23], and Cassatella-Contra and Ma˜nas [10], as

a generalization of the classical result of Fokas, Its and Kitaev [21]. This formulation

has been used in several examples, like Hermite and Laguerre–type MVOPs in [7, 8] or

matrix biorthogonal polynomials in [5, 6], in order to obtain algebraic and diﬀerential

identities for MVOPs that can be seen as non-commutative analogues of well known

identities in the theory of integrable systems, such as the Toda lattice equation or

Painlev´e equations.

Asymptotic results for MVOPs obtained from the Riemann–Hilbert formulation

using the Deift–Zhou method [13] of steepest descent are much more scarce. In the

last few years, MVOPs have appeared in the area of integrable probability, more

precisely in the study of random tilings of plane ﬁgures. We mention the recent work

by Duits and Kuijlaars [17] and Berggren and Duits [3] on periodic tilings of the Aztec

diamond, as well as the papers by Charlier [11] and by Groot and Kuijlaars [22] on

doubly periodic lozenge tilings of a hexagon. In these cases, an essential step in the

asymptotic analysis is the connection between matrix orthogonality in the complex

plane and scalar orthogonality on suitable curves in a Riemann surface.

Our results are strong asymptotic formulas for Pn(z) as n→ ∞, for zin three

regions in the complex plane, namely in the exterior region C\[−1,1], in the oscillatory

region (−1,1) away from the endpoints, and near the endpoints. An important aspect

of this work is the fact that we use diﬀerent factorizations of the weight matrix for

the asymptotic analysis: in the outer region and on the interval (−1,1), we use a

matrix Szeg˝o function D, which is obtained from a matrix spectral factorization of

the weight on the unit circle; in neighborhoods of the endpoints, we use the spectral

decomposition of W(x), since the possible vanishing of the eigenvalues at z=±1 is

essential in the construction of the local parametrices. The same methodology allows

us to include asymptotic expansions for the recurrence coeﬃcients as well.

Throughout we assume that Wis of the form (1.1) with Hsatisfying Assumption

1.1, and Pnis the degree nmonic MVOP satisfying (1.2). We use ATto denote the

transpose of a matrix Aand A∗for its Hermitian transpose. For a matrix valued

function A(z) deﬁned for z∈C\Σ, where Σ is an oriented contour, we use A+(x)

(A−(x)) for the limits of A(z) as z→x∈Σ from the +-side (−-side). The +-side

(−-side) is on our left (right) as we follow Σ according to its orientation.

1.2 Factorizations of the weight matrix

Our asymptotic results rely on three factorizations of the weight matrix.

1.2.1 First factorization

The ﬁrst one is the familiar spectral decomposition of H(x)

H(x) = Q(x)Λ(x)Q(x)∗, x ∈[−1,1] (1.3)

with a unitary matrix Q(x) and a diagonal matrix

Λ(x) = diag (λ1(x), . . . , λr(x)) (1.4)

containing the eigenvalues λj(x), j= 1, . . . , r of H(x). The assumption that His real

analytic on [−1,1] has the following important consequence.

Lemma 1.2. Q(x)and Λ(x)can (and will) be taken to be real analytic on [−1,1].

Proof. This is a well-known theorem of Rellich, see [40] or [38, Theorem 1.4.4].

We choose Q(x) and Λ(x) as in Lemma 1.2, and we continue to use Qand Λ for

their analytic continuations to a neighborhood of [−1,1] in the complex plane. Then

each eigenvalue λj(x), j= 1, . . . , r is analytic in that same neighborhood of [−1,1],

and it satisﬁes λj(x)>0 for x∈(−1,1) because of Assumption 1.1 (b), but λj(x)

could be zero at x=±1, since we do not assume positive deﬁniteness of Hat the

endpoints.

Deﬁnition 1.3. We deﬁne for j= 1, . . . , r,

(a) njis the order of vanishing of λj(x) at x= 1, where we put nj= 0 if λj(1) >0,

and

αj=α+nj,(1.5)

(b) mjis the order of vanishing of λj(x) at x=−1, where we put mj= 0 if

λj(−1) >0, and

βj=β+mj.(1.6)

Because of Assumption 1.1 (d) at least one of the numbers n1, . . . , nris equal to

zero, and similarly for the mj’s. Thus we have

min{nj|j= 1, . . . , r}= min{mj|j= 1, . . . , r}= 0.

We emphasize that λj(x), for j= 1, . . . , r are the eigenvalues of H(x), and so by (1.1)

the eigenvalues of W(x) are (1 −x)α(1 + x)βλj(x) for j= 1, . . . , r.

1.2.2 Second factorization

The second factorization of W(x) is less familiar.

Proposition 1.4. There exists an analytic matrix valued function D:C\[−1,1] →

Cr×rwith boundary values D±on (−1,1) satisfying

W(x) = D−(x)D−(x)∗=D+(x)D+(x)∗,(1.7)

where D(z)is invertible for every z∈C\[−1,1], and such that

D(∞) = lim

z→∞ D(z) (1.8)

exists and is invertible as well.

Proposition 1.4 follows from Lemma 3.2 below.

A similar factorization, but for weight matrices on the unit circle appeared in

[3], in the study of correlation functions for determinantal processes involving inﬁnite

Toeplitz minors, which arise in random tilings of certain planar domains.

Remark 1.5. We consider D(z) as a matrix valued Szeg˝o function. It arises from a

matrix spectral factorization of the weight matrix W. It is unique up to a constant

unitary matrix. That is, if Dsatisﬁes the conditions of Proposition 1.4 and Uis a

unitary matrix, independent of z, then DU satisﬁes the conditions as well. Uniqueness

of the matrix valued Szeg˝o function is guaranteed if we require that D(∞) is a positive

deﬁnite Hermitian matrix. We call this the normalized matrix valued Szeg˝o function.

If W(x) is real valued for x∈(−1,1), then the normalized matrix valued Szeg˝o

function Dwill satisfy the symmetry condition

D(z) = D(z), z ∈C\[−1,1].(1.9)

In that case D−(x) = D+(x) and the factorization (1.7) can be alternatively written

W(x) = D−(x)D+(x)T=D+(x)D−(x)T.(1.10)

Also D(∞) is a positive deﬁnite real matrix in this case.

1.2.3 Third factorization

The third factorization is very much related to the spectral decomposition (1.3). We

use modiﬁed eigenvalues

λj= (−1)njλj, j = 1, . . . , r, (1.11)

and e

Λ = diag(e

λ1,...,e

λr).(1.12)

Recall from Deﬁnition 1.3 that njdenotes the order of vanishing of λjat x= 1. Thus

λj(x)>0 for x∈(1,1 + δ) for some δ > 0, and we use e

λj(x)1/2to denote its positive

square root. This has an analytic continuation to a neighborhood of [−1,1] with a

branch cut along (−∞,1] that we also denote by e

λ1/2

j. Then we deﬁne

Λ1/2= diag e

λ1/2

1,...,e

λ1/2

r,(1.13)

and

V(z) = (z−1)α/2(z+ 1)β/2Q(z)e

Λ(z)1/2.(1.14)

which is deﬁned and analytic with a branch cut along (−∞,1]. In particular it is

deﬁned and analytic in D(1, δ)\(1 −δ, 1] for some δ > 0.

We will use Vfor the local analysis around 1. Near −1 we have a similarly deﬁned

matrix valued function. We deﬁne

Λ = diag b

λ1,...,b

λr,b

λj= (−1)mjλj,(1.15)

so that b

λj(x)>0 for x∈(−1−δ, −1) for some δ > 0. Then we deﬁne

V(z) = (1 −z)α/2(−1−z)β/2Q(z)b

Λ1/2(z),(1.16)

deﬁned with a branch cut along [−1,∞).

The third factorization of Wis as follows:

Lemma 1.6. We have for x∈(−1,1),

W(x) = V−(x)V−(x)∗=V+(x)V+(x)∗

V−(x)b

V−(x)∗=b

V+(x)b

V+(x)∗,(1.17)

where Vand b

Vare deﬁned by (1.14) and (1.16).

Proof. This follows by straightforward calculation from the deﬁnitions (1.14) and

(1.16). See also Lemma 3.4 for details.

Comparing (1.17) and (1.7) we see that Vand b

Vshare the same factorization

property with the matrix valued Szeg˝o function D. Actually D±(x)−1V±(x) and

D±(x)−1b

V±(x) are unitary matrices for every x∈(−1,1), see formula (3.46) below.

For our asymptotic results we need their values at the endpoints.

Lemma 1.7. The two limits

U1= lim

z→1D(z)−1V(z), U−1= lim

z→−1D(z)−1b

V(z) (1.18)

exist, and deﬁne unitary matrices U1and U−1.

The proof of Lemma 1.7 is in Section 3.5.8.

1.3 Asymptotics in the exterior region

Throughout the paper, we need the conformal map

ϕ(z) = z+ (z2−1)1/2, z ∈C\[−1,1] (1.19)

from C\[−1,1] to the exterior of the unit circle. Our ﬁrst result is the asymptotics of

Pn(z) as n→ ∞ for z∈C\[−1,1]. The main term in the asymptotic formula (1.20) is

not new as it is known at least since [2], where it is proved under weaker assumptions

as well, namely Wis assumed to satisfy a matrix Szeg˝o condition on [−1,1] with a

ﬁnite number of mass points outside [−1,1]). See also [30] for an inﬁnite number of

mass points.

Theorem 1.8. Let Wbe the weight matrix (1.1) with Hsatisfying Assumption 1.1.

Let Dbe the matrix Szeg˝o function associated with Was in Proposition 1.4. Then as

n→ ∞ the monic MVOP Pnhas an asymptotic series expansion

2nPn(z)

ϕ(z)n∼ϕ(z)1/2

√2(z2−1)1/4D(∞)"Ir+∞

k=1

Πk(z)

nk#D(z)−1, z ∈C\[−1,1],(1.20)

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Asymptoticsofmatrixvaluedorthogonalpolynomialson[1;1]AlfredoDea~no*1,ArnoB.J.Kuijlaars2,andPabloRoman31DepartmentofMathematics,UniversidadCarlosIIIdeMadrid,Spain2DepartmentofMathematics,KatholiekeUniversiteitLeuven,Leuven,Belgium3FaMAF-CIEM,UniversidadNacionaldeCordoba,ArgentinaApril11,2023Abstr...

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Asymptotics of matrix valued orthogonal polynomials on 11 Alfredo Dea no1 Arno B.J. Kuijlaars2 and Pablo Rom an3

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