Diffraction of plane waves, spherical waves, and beyond

TL;DR

This paper extends diffraction theory to unbounded and spherical wave settings, introduces radial almost periodicity, and provides explicit formulas and new proofs for diffraction phenomena.

Contribution

It develops a framework for diffraction of unbounded and spherical waves, introducing radial almost periodicity and connecting it to Besicovitch functions.

Findings

Diffraction of plane waves linked to Besicovitch almost periodic functions.

Spherical wave diffraction results in a single sphere.

Established a radial analogue of Lebesgue decomposition.

Abstract

We review the diffraction theory for plane waves and establish its connection to the diffraction of Besicovitch almost periodic functions, extending the theory to an unbounded setting and providing explicit formulas. Then, we give an alternative proof that the diffraction of a spherical wave in $\mathbb{R}^d$ is a single sphere, which was recently shown in \cite{BKM25}. After developing a suitable framework for working with spherically symmetric measures, including a radial analogue of the usual Lebesgue decomposition, we introduce the notion of radial almost periodicity. In particular, we define a space of Besicovitch radially almost periodic functions and show that this space contains precisely the functions whose radial part is Besicovitch almost periodic. The paper concludes with a diffraction analysis of these functions.