RT Journal Article
T1 Asymptotics of matrix valued orthogonal polynomials on [−1,1]
A1 Deaño Cabrera, Alfredo
A1 Kuijlaars, Arno B.J.
A1 Román, P.
AB We analyze the large degree asymptotic behavior of matrix valued orthogonal polynomials (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 Szegő function. We illustrate the results with two main examples, MVOPs of Jacobi and Gegenbauer type, coming from group theory.
PB Elsevier Inc
SN 0001-8708
YR 2023
FD 2023-06-15
LK https://hdl.handle.net/10016/39150
UL https://hdl.handle.net/10016/39150
LA eng
NO Comunidad de Madrid (Spain). CM/JIN/2021-014
NO acknowledges financial support from Dirección General de Investigación e Innovación, Consejería de Educación e Investigación of Comunidad de Madrid (Spain),and Universidad de Alcalá under grant CM/JIN/2021-014, and Comunidad de Madrid(Spain) under the Multiannual Agreement with UC3M in the line of Excellence ofUniversity Professors (EPUC3M23), and in the context of the V PRICIT (RegionalProgramme of Research and Technological Innovation). Research supported by GrantPID2021-123969NB-I00, funded by MCIN/AEI/10.13039/501100011033, and by grantPID2021-122154NB-I00 from Spanish Agencia Estatal de Investigación.A. D. acknowledges financial support and hospitality from IMAPP, Radboud Universiteit Nijmegen, and in particular Prof. Erik Koelink, during a visit to Nijmegen in June2022.
NO Gobierno de España. MCIN/AEI/10.13039/501100011033
NO Gobierno de España. PID2021-122154NB-I00
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RD 17 jul. 2024