Association between temperate distributions and analytical functions in the context of wave-front sets

TL;DR

This paper establishes a connection between temperate distributions, analytic functions in convex tubes, and wave-front sets within Fourier Banach spaces, providing a new perspective on the structure of singularities.

Contribution

It introduces a novel association between temperate distributions and analytic functions in convex tubes, linking wave-front sets to Fourier BF-space membership.

Findings

Wave-front set points correspond to non-membership in Fourier BF-spaces.

Every temperate distribution can be associated with an analytic function in a convex tube.

The characterization provides a new analytical tool for studying wave-front sets.

Abstract

Let B be a translation invariant Banach function space (BF-space). In this paper we prove that every temperate distribution f can be associated with a function F analytic in the convex tube Omega={z in C^d; |Im z|<1} such that the wave-front set of f of Fourier BF-space types in intersection with R^d \times S^{d-1} consists of the points (x,\xi) such that F does not belong to the Fourier BF-space at x-i\xi.