Analytical study of the pantograph equation using Jacobi theta functions

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Approximation Theory Pub Date : 2023-09-24 DOI:10.1016/j.jat.2023.105974

Changgui Zhang

引用次数: 0

Abstract

The aim of this paper is to use the analytic theory of linear $q$ -difference equations for the study of the functional-differential equation $y^{'} (x) = a y (q x) + b y (x)$ , where $a$ and $b$ are two non-zero real or complex numbers. When $0 < q < 1$ and $y (0) = 1$ , the associated Cauchy problem admits a unique power series solution, $\sum_{n \geq 0} \frac{{(- a / b; q)}_{n}}{n!} {(b x)}^{n}$ , that converges in the whole complex $x$ -plane. The principal result obtained in the paper explains how to express this entire function solution into a linear combination of solutions at infinity with the help of integral representations involving Jacobi theta functions. As a by-product, this connection formula between zero and infinity allows one to rediscover the classic theorem of Kato and McLeod on the asymptotic behavior of the solutions over the real axis.

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受电弓方程的Jacobi-theta函数分析研究

本文的目的是利用线性q-差分方程的解析理论研究函数微分方程y′（x）=ay（qx）+by（x），其中a和b是两个非零实数或复数。当0<；q<；1和y（0）=1时，相关的Cauchy问题得到一个唯一的幂级数解，∑n≥0（−a/b；q）nn！（bx）n，其在整个复x平面上收敛。本文获得的主要结果解释了如何在涉及Jacobiθ函数的积分表示的帮助下，将整个函数解表示为无穷远处解的线性组合。作为副产品，这个零和无穷大之间的联系公式允许我们重新发现Kato和McLeod关于实轴上解的渐近行为的经典定理。

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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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.