SOME SPECIAL TAUBERIAN-THEOREMS FOR FOURIER TYPE INTEGRALS AND EIGENFUNCTION-EXPANSIONS OF STURM-LIOUVILLE EQUATIONS ON THE WHOLE LINE

We offer a new proof of a special Tauberian theorem for Fourier type i ntegrals. This Tauberian theorem was already considered by us in the p apers [1] and [2]. The idea of our initial proof was simple, but the d etails were complicated because we used Bochner's definition of genera lized Fourier transform for functions of polynomial growth. In the pre sent paper we work with L. Schwartz's generalization. This leads to si gnificant simplification. The paper consists of six sections. In Secti on 1 we establish an integral representation of functions of polynomia l growth (subjected to some Tauberian conditions), in Section 2 we pro ve our main Tauberian theorems (Theorems 2.1 and 2.2.), using the inte gral representation of Section 1, in Section 3 we study the asymptotic behavior of M. Riesz's means of functions of polynomial growth, in Se ctions 3 and 5 we apply our Tauberian theorems to the problem of equic onvergence of eigenfunction expansions of Sturm-Liouville equations an d expansion in ordinary Fourier integrals, and in Section 6 we compare our general equiconvergence theorems of Sections 4 and 5 with the wel l known theorems on eigenfunction expansions in classical orthogonal p olynomials. In some sense this paper is a re-made survey of our result s obtained during the period 1953 - 58. Another proof of our Tauberian theorem and some generalization can be found in the papers [3] and [4 ].