Спектральный метод В.А. Ильина. Несамосопряженные операторы. II. Оценки скорости равносходимости спектральных разложений

Abstract

This monograph continues the series devoted to the results obtained with the help of V.A. Ilyin spectral method for studying differential operators. The problem of obtaining estimates for the rate of equiconvergence and estimates for the rate of convergence of spectral expansions of functions over systems of root functions of ordinary differential operators of various orders given on a finite interval of the real line or on the entire line is investigated. Convergence and equiconvergence of spectral expansions of functions is considered in various metrics. Equiconvergence theorems make it possible to transfer known results on the convergence or divergence of well-studied series (for example, trigonometric series or series in exponential systems) into spectral expansions in terms of eigenfunctions and associated functions of differential operators. The first theorems of equiconvergence of spectral expansions of functions - the Steklov- Hobson-Haar and Tamarkin-Stone theorems - are given. We present and prove in detail the first theorem containing an estimate for the rate of local equiconvergence of spectral expansions of functions - the Ilyin-Yo theorem, as well as the first theorem containing an estimate of the rate of equiconvergence of spectral expansions of functions on the entire interval. A theorem is formulated and proved that generalizes the classical theorem of F. Riesz (Riesz- Fischer) to biorthogonal systems of functions. This theorem, along with local and non-local mean value formulas, which are given much attention in the book, served as a tool that made it possible to extend the results on estimates of the rate of local equiconvergence to the case of equiconvergence over the entire segment of the operator specification. The book is intended for mathematicians, physicists, applied mathematicians and engineers who are in contact with the spectral theory of differential operators, students and graduate students of mathematical specialties of universities.