The Fourier transform of vector-valued functions

Abstract

For each natural numbern, letun(x)=(1—cosnx)/πnx2(xɛℝ). It is well–known that a bounded continuous functionfon the real line ℝ is the Fourier transform of an integrable function on ℝ if and only if the functions Φn(f) (n= 1, 2,…), defined byform a Cauchy sequence in the spaceL1(ℝ) (cf. [2]). Such a characterization, which can be extended to functions defined on a locally compact Abelian group more general than ℝ, is based on the fact that the spaceL1(ℝ) is complete with respect to convergence in mean.