A Sequential Approach to Mild Distributions

Abstract

We describe an elementary sequential realization of the Banach Gelfand triple (S0(R^d), L2(R^d), S0'(R^d)). Here S0(R^d) is a Segal algebra of test functions, L2(R^d) is the usual Hilbert space, and S0'(R^d) is its dual space of mild distributions. This framework is fundamental for Gabor analysis and provides a natural setting for the generalized Fourier transform and the short-time Fourier transform. Inspired by Lighthill's sequential approach to tempered distributions, we construct an extended domain for the short-time Fourier transform from equivalence classes of extended mild Cauchy sequences, abbreviated as ECmiCS. Their representatives are sequences of bounded continuous functions. The construction avoids Lebesgue integration and the theory of tempered distributions. Our main result identifies the resulting sequential space canonically with S0'(R^d), thereby recovering the Banach Gelfand triple in an elementary form.