A regularization method for a class of dual series equations
type: Item
instance of: scholarly article
zbMATH Open document ID: 0999.30026
DOI: 10.1023/A:1015283624714
author: A. V. Brovenko
author: P. N. Melezhik
author: A. E. Poedinchuk
publication date: 2002-04-24T00:00:00Z
Mathematics Subject Classification ID: 30E25
Mathematics Subject Classification ID: 42A16
Mathematics Subject Classification ID: 34M50
zbMATH DE Number: 1733113
zbMATH Keywords: Fourier expansions
zbMATH Keywords: Riemann-Hilbert problem
zbMATH Keywords: operator equation
MaRDI profile type: Publication
full work available at URL: a:1015283624714
OpenAlex ID: W4165735
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published in: Ukraïns'kyĭ Matematychnyĭ Zhurnal
title: A regularization method for a class of dual series equations
review text: The purpose of this paper is to construct a method of regularization to the dual series equations \(\sum_{n=-\infty}^{+\infty}n x_n e^{i n \varphi} - a\sum_{n=-\infty}^{+\infty}n x_n e^{i n \varphi} = \sum_{n=-\infty}^{+\infty} g_n e^{i n \varphi},\) \(|\varphi|<\theta,\) where \(g_n = f_n+\sum_{m=-\infty}^{+\infty}V_{nm} x_m,\) \(x_n\in l_2(1) = \{x = (x_n)_{n=-\infty}^{+\infty}: \sum_{n=-\infty}^{+\infty}|x_n|^2(|n|+1)<\infty\}.\) And there is additionally supposed that \( f = (f_n)_{n=-\infty}^{+\infty} \in l_2(1), \) the matrix operator \( V: l_2 \to l_2 \) is bounded, series \(\sum_{n=-\infty}^{+\infty} x_n e^{i n \varphi}\) are the Fourier expansions for a function from \( L_2 [-\pi, \pi],\) and \(\sum_{n = -\infty}^{-1} n x_n e^{i n \varphi},\) \(\sum_{n=1}^{\infty} n x_n e^{i n \varphi},\) \(\sum_{n=-\infty}^{+\infty} g_n e^{i n \varphi} \) are the Fourier expansions for a function from \( L_1[-\pi, \pi].\) NEWLINENEWLINENEWLINEThe main idea of the paper is to reduce the original problem to the Riemann-Hilbert problem for an unclosed contour of the unit circle. So the original problem is reduced to the operator equation \(x+Hx=h,\) where \(H=AV:l_2\to l_2, \) is a compact operator and \(h=Af\).
description: scientific article; zbMATH DE number 1733113