Some inequalities for Fourier transforms

Abstract

(5) F(2) = 0(0+) This expression follows readily from a known theorem on Fourier transforms [1, Theorem 3, p. 13] if +(t) is defined in the interval (-x , 0) by the relation (-t) =c(t). In this paper an inequality on f(x) for monotonic decreasing +(t) and inequalities on g(x) and F(x) for non-negative +(t) are obtained when the bounds on these functions are known for x 2 1. The result on f(x) is similar to an inequality obtained by Boas and Kac [2]. In the more general case when 4(t) 2 0, they use the more restrictive condition, f(x) = 0 for x 2 1, to show that then