On a general Tauberian theorem

Abstract

In this article the celebrated Tauberian theorem by Norbert Wiener ([1], [2]) is considerably extended by the introduction of a wider class of translation kernels. A number of applications are given to show the fruitfulness of this new generalization. Introduction. To begin with we quote the celebrated Tauberian theorem by Norbert Weiner ([1], [2]). Assume that K(x) eLI(-co, oo), 9(x) eL(-oo, oo) and let the Fourier transform of K be 00 for any real argument. The relation (1) lim 9 ()K(x ) dd = A K($) dt, X= 0 -% _oo where A is a number, implies for any H(x) e L1(oc, co) (2) lim fJ()H(x t) d$ = A H(t) d$. SC= C_t, -oo -oo It was also shown in the quoted paper that a large number of classical Tauberian theorems can be reduced to the result stated above concerning the two integrals. The scope of Wiener's theorem can, however, be considerably extended by the introduction of a wider class of translation kernels. Let v(x) be a positive measurable function on the real axis satisfying the conditions: