Functions Whose Fourier Transform Decays at Infinity; An Extension of the Riemann Lebesgue Lemma

摘要

An extension of the Riemann Lebesgue Lemma is stated and proved. The class of functions considered are locally L sub 1 on (0, infinity) and have asymptotic expansion as t approaches infinity of the form f(t) is approximately equal to exp(i alpha (t sup nu)). Summation, n=0 to infinity, summation m=0 to N(m) of ((C sub mn)(t sup(r sub m))((log t) to the nth power). Here the sequence (Re r sub m) increases monotonically to plus infinity, Re r sub 0 > 0 and N(m) is finite for each m. (Author)