On asymptotics of Fourier transform for functions of certain classes

Abstract

In many cases it is important to know the asymptotic behaviour of Fourier transform. In the article this problem is solved for several classes of functions, which were not examined before (except perhaps in the recent paper by TRmtrB [31]). The relations defining the terms of either class affiliation are mainly the integral analogues of known conditions for the integrability of trigonometric series. The main result Theorem 2 has the Boas-Teljakovskii conditions [24, 25] as its prototype. Besides, the corresponding analogues of the FOMIN [I 1], (Sidon-)Tt~LJAKOVSKII [26] conditions are obtained. The mentioned conditions for integrability of trigonometric series are deduced from these results in a somewhat stronger form. The generalizations to the multiple case are obtained: the multidimensional analogue of Theorem 2 is proved and the ways for getting other asymptotic formulae as well as slightly stronger versions of conditions for integrability of trigonometric series obtained earlier by NOSENKO, TELJAKOVSKII, ZADEREI [18--20, 27, 32--34] are also pointed out.