Some Special Tauberian Theorems for Fourier Type Integrals and Eigenfunction Expansions of Sturm-Liouville Equations on the Whole Line
✍ Scribed by B. M. Levitan
- Publisher
- John Wiley and Sons
- Year
- 1995
- Tongue
- English
- Weight
- 598 KB
- Volume
- 176
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
We offer a new proof of a special Tauberian theorem for Fourier type integrals. This Tauberian theorem was already considered by us in the papers [1] and [2]. The idea of our initial proof was simple, but the details were complicated because we used Bochner's definition of generalized Fourier transform for functions of polynomial growth. In the present paper we work with L. Schwartz's generalization. This leads to significant simplification. The paper consists of six sections. In Section 1 we establish an integral representation of functions of polynomial growth (subjected to some Tauberian conditions), in Section 2 we prove our main Tauberian theorems (Theorems 2.1 and 2.2.), using the integral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Sections 4 and 5 we apply our Tauberian theorems to the problem of equiconvergence of eigenfunction expansions of Sturm‐Liouville equations and expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the well known theorems on eigenfunction expansions in classical orthogonal polynomials. In some sense this paper is a re‐made survey of our results obtained during the period 1953‐58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4].