Sobolev Orthogonal Polynomials on the unit circle with varying inner product
Abstract
In this work, we investigate the validity of the Szegö type strong asymptotic formula for the Sobolev orthogonal polynomials associated with a Sobolev-type inner product depending on a parameter s, which is defined as ⟨ f,g⟩ λ/ns:= ∫𝕋 f(z)g(z)̅ dμ1(z)+ λ/ns ∫𝕋 f′(z)g′(z)̅ dμ2(z), where $s$ is fixed and s≥ 2, and λ>0. Here, μ1, μ2 are absolutely continuous with respect to the normalized lebesgue measure and ∫𝕋 μ′i(z) dm=1, where μi′(z)= dμi/dm >0 for i=1,2. Further, μ1, μ2 belongs to the Szegö class and also has non-zero bounded Radon-Nikodym derivatives on the unit circle. %satisfies $ 0 <m \leq \dfrac{μ^{\prime}_1(e^{ιθ})}{μ^{\prime}_2(e^{ιθ})}\leq M < \infty $. We observe that the measure appearing in the Szegö function of the strong asymptotic limit depends on s. As a specific case, we consider the Bernstein-Christoffel Sobolev orthogonal polynomials and study their properties, including the asymptotic distribution and location of their zeros for different values of s. We further establish a spectral characterization of their zeros as the eigenvalue of an explicitly constructed modification of the associated Hessenberg matrix and provide numerical evidence supporting the obtained asymptotic and zeros results.
Related
- A reciprocal-Gamma Mehler--Heine limit for Krawtchouk--Sobolev type orthogonal polynomials with an exterior mass point
- Orthogonal Pairs in Maps from the Sphere to the Circle
- Matrix orthogonal polynomials related to diffference operators for L-orthogonal polynomials
- Identities and congruences involving orthogonal polynomials and Apéry-like numbers
- An Exploration of Lacunary Orthogonal Polynomials Associated with the Geronimus Transformation
BibTeX
@misc{majumder2026sobolev,
title = {{Sobolev Orthogonal Polynomials on the unit circle with varying inner product}},
author = {Sourav Majumder and A. Swaminathan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31994},
eprint = {2610.31994},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31994},
}
RIS
TY - UNPB
TI - Sobolev Orthogonal Polynomials on the unit circle with varying inner product
AU - Sourav Majumder
AU - A. Swaminathan
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31994
UR - https://arxiv.org/abs/2610.31994
AB - In this work, we investigate the validity of the Szegö type strong asymptotic formula for the Sobolev orthogonal polynomials associated with a Sobolev-type inner product depending on a parameter s, which is defined as ⟨ f,g⟩ _λ/n^s:= ∫_𝕋 f(z)g(z)̅ dμ_1(z)+ λ/n^s ∫_𝕋 f^′(z)g^′(z)̅ dμ_2(z), where $s$ is fixed and s≥ 2, and λ>0. Here, μ_1, μ_2 are absolutely continuous with respect to the normalized lebesgue measure and ∫_𝕋 μ^′_i(z) dm=1, where μ_i^′(z)= dμ_i/dm >0 for i=1,2. Further, μ_1, μ_2 belongs to the Szegö class and also has non-zero bounded Radon-Nikodym derivatives on the unit circle. %satisfies $ 0 <m \leq \dfrac{μ^{\prime}_1(e^{ιθ})}{μ^{\prime}_2(e^{ιθ})}\leq M < \infty $. We observe that the measure appearing in the Szegö function of the strong asymptotic limit depends on s. As a specific case, we consider the Bernstein-Christoffel Sobolev orthogonal polynomials and study their properties, including the asymptotic distribution and location of their zeros for different values of s. We further establish a spectral characterization of their zeros as the eigenvalue of an explicitly constructed modification of the associated Hessenberg matrix and provide numerical evidence supporting the obtained asymptotic and zeros results.
ER -