Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier transform

TL;DR

This paper characterizes ultradifferentiable functions with compact support, defined by weight matrices, through the decay of their Fourier transforms, providing new insights into their structure and non-quasianalyticity.

Contribution

It introduces a novel technique for characterizing ultradifferentiable functions via Fourier decay using multi-index matrices and analyzes their stability and non-quasianalyticity.

Findings

Functions with compact support are characterized by Fourier decay properties.

The construction of multi-index matrices is stable and useful for analysis.

Non-quasianalyticity of classes is fully characterized.

Abstract

We prove that functions with compact support in non-quasianalytic classes of Roumieu-type and of Beurling-type defined by a weight matrix with some mild regularity conditions can be characterized by the decay properties of their Fourier transform. For this we introduce the abstract technique of constructing from the original matrix multi-index matrices and associated function spaces. We study the behaviour of this construction in detail and characterize its stability. Moreover non-quasianalyticity of the classes is characterized.