Harmonic Analysis on the Space of M-positive Vectors

TL;DR

This paper introduces a new harmonic analysis framework on the space of M-positive vectors, defining Fourier transforms and classical harmonic analysis results analogous to Walsh analysis.

Contribution

It develops a novel harmonic analysis on M-positive vector spaces, including Fourier transform, Poisson summation, and Plancherel theorem, extending Walsh analysis concepts.

Findings

Established Fourier transform and harmonic analysis tools for M-positive vector spaces.

Proved Poisson summation and Plancherel theorem analogues.

Characterized a Schwartz-like class with compactly supported functions and transforms.

Abstract

Given a dilation matrix M, a so-called space of M-positive vectors in the Euclidean space is introduced and studied. An algebraic structure of this space is similar to the positive half-line equipped with the termwise addition modulo 2, which is used in the Walsh analysis. The role of harmonics is played by some analogues of the classical Walsh functions. The concept of Fourier transform is introduced, and the Poisson summation formula, Plancherel theorem, Vilenkin-Chrestenson formulas and so on are proved. A kind of analogue of the Schwartz class is studied. This class consists of functions such that both the function itself and its Fourier transform have compact support.