Problems of localization and convergence for Fourier series in fundamental systems of functions of the Laplace operator

V. A. Il'in

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 23 (1968) no. 2, pp. 59-116

Résumé

The paper deals with the problems of localization and convergence of Fourier series with respect to a so-called $fundamental system of the Laplace operator$. (As understood by the author in this paper, a fundamental system of functions includes the eigenfunctions of all boundary-value problems and is characterized by the absence of any kind of boundary conditions). Chapter 1 of the paper contains a survey of all the important results on the problems of localization and convergence of Fourier series, both for concrete systems of eigenfunctions of the Laplace operator (and, in particular, for the multiple trigonometric system) and for arbitrary fundamental systems of functions of this operator. In Chapters 2–5 detailed proofs are given for the recent results of the author concerning general fundamental systems of functions of the Laplace operator, including: 1) a comprehensive solution of the localization problem for an arbitrary $N$-dimensional domain in the Sobolev classes $W_2^\alpha$ (with non-integral $\alpha$), 2) a comprehensive solution of the localization and convergence problem for an arbitrary odd-dimensional domain in the Hölder classes $C^{(n,\alpha)}$, 3) almost definitive conditions for localization and convergence for an arbitrary even-dimensional domain, 4) a proof of the result that in the class of all $N$-dimensional domains the smoothness conditions prescribed for the function $f(x)$ to be expanded are best possible even for an arbitrary rearrangement of the terms of the Fourier series.