Diffraction theory and almost periodic distributions

TL;DR

This paper develops a comprehensive framework for almost periodic distributions, extending classical concepts to tempered distributions, and establishes connections with Fourier analysis and measure theory.

Contribution

It introduces new notions of almost periodicity for tempered distributions and links Eberlein decomposition with Fourier duality, expanding the theoretical understanding.

Findings

Weakly almost periodic distributions satisfy Eberlein decomposition.

Tempered distributions with measure Fourier transform are weakly almost periodic.

Eberlein decomposition corresponds to Lebesgue decomposition via Fourier duality.

Abstract

We introduce and study the notions of translation bounded tempered distributions, and autocorrelation for a tempered distrubution. We further introduce the spaces of weakly, strongly and null weakly almost periodic tempered distributions and show that for weakly almost periodic tempered distributions the Eberlein decomposition holds. For translation bounded measures all these notions coincide with the classical ones. We show that tempered distributions with measure Fourier transform are weakly almost periodic and that for this class, the Eberlein decomposition is exactly the Fourier dual of the Lesbegue decomposition, with the Fourier-Bohr coefficients specifying the pure point part of the Fourier transform. We complete the project by looking at few interesting examples.