广义贝塞尔多项式单位圆上的正交权重

IF 0.9 3区数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2024-10-22 DOI:10.1016/j.jat.2024.106115

Sergey M. Zagorodnyuk

引用次数: 0

摘要

本文研究广义贝塞尔多项式 yn(x,a,b)（用 Krall 和 Frink 的符号表示）。设 a>1, b∈(-1/3,1/3)∖{0}。在这种情况下，我们在单位圆上为 yn(x,a,b) 提出以下正连续权值 p(θ)=p(θ,a,b) ：2πp(θ,a,b)=-1+2(a-1)∫01e-bucosθcos(businθ)(1-u)a-2du，其中θ∈[0,2π]。即，我们有∫02πyn(eiθ,a,b)ym(eiθ,a,b)p(θ,a,b)dθ=Cnδn,m,Cn≠0,n,m=0,1,2,....。注意，这个正交性与单位圆上正交多项式的通常正交性不同。给出了上述正交性的一些应用。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

In this paper we study the generalized Bessel polynomials

y_{n} (x, a, b)

(in the notation of Krall and Frink). Let

a > 1

b \in (- 1 / 3, 1 / 3) ∖ {0}

. In this case we present the following positive continuous weights

p (θ) = p (θ, a, b)

on the unit circle for

y_{n} (x, a, b)

2 π p (θ, a, b) = - 1 + 2 (a - 1) \int_{0}^{1} e^{- b u cos θ} cos (b u sin θ) {(1 - u)}^{a - 2} d u,

where

θ \in [0, 2 π]

. Namely, we have

\int_{0}^{2 π} y_{n} (e^{i θ}, a, b) y_{m} (e^{i θ}, a, b) p (θ, a, b) d θ = C_{n} δ_{n, m}, C_{n} \neq 0, n, m = 0, 1, 2, \dots .

Notice that this orthogonality differs from the usual one for orthogonal polynomials on the unit circle. Some applications of the above orthogonality are given.

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来源期刊

Journal of Approximation Theory 数学-数学

CiteScore

1.90

自引率

11.10%

发文量

审稿时长

6-12 weeks

期刊介绍： The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.

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