# Symmetry problems in harmonic analysis

## TL;DR

This paper addresses symmetry problems in harmonic analysis, proving that certain integral conditions imply geometric symmetry of domains, with results related to the refined Schiffer's conjecture and spherical symmetry.

## Contribution

The paper formulates and solves symmetry problems in harmonic analysis, establishing conditions under which domains must be spherical, and links these to the refined Schiffer's conjecture.

## Key findings

- Integral conditions imply spherical symmetry of domains.
- Connected bounded domains with certain harmonic properties are necessarily balls.
- Results confirm conjectures relating harmonic integrals to domain symmetry.

## Abstract

Symmetry problems in harmonic analysis are formulated and solved. One of these problems is equivalent to the refined Schiffer's conjecture which was recently proved by the author. Let $k=const>0$ be fixed, $S^2$ be the unit sphere in $\mathbb{R}^3$, $D$ be a connected bounded domain with $C^2-$smooth boundary $S$, $j_0(r)$ be the spherical Bessel function. The harmonic analysis symmetry problems are stated in the following theorems: {\bf Theorem A.} {\em Assume that $\int_S e^{ik\beta \cdot s}ds=0$ for all $\beta\in S^2$. Then $S$ is a sphere of radius $a$, where $j_0(ka)=0$. } {\bf Theorem B.} {\em Assume that $\int_D e^{ik\beta \cdot x}dx=0$ for all $\beta\in S^2$. Then $D$ is a ball.