Integrals of spherical harmonics with Fourier exponents in multidimensions

TL;DR

This paper derives explicit formulas for integrals of spherical harmonics with Fourier exponents on spheres, which are relevant in Radon transforms and electromagnetic theory, providing exact constants for specific harmonic classes.

Contribution

It provides analytic formulas for these integrals, including explicit constants for harmonics related to Radon transforms, and suggests methods for general cases.

Findings

Exact formulas for integrals of spherical harmonics with Fourier exponents.

Explicit constants for harmonics in Radon transform applications.

Proposed methods for calculating constants in general cases.

Abstract

We consider integrals of spherical harmonics with Fourier exponents on the sphere $S^n ,\, n \geq 1$. Such transforms arise in the framework of the theory of weighted Radon transforms and vector diffraction in electromagnetic fields theory. We give analytic formulas for these integrals, which are exact up to multiplicative constants. These constants depend on choice of basis on the sphere. In addition, we find these constants explicitly for the class of harmonics arising in the framework of the theory of weighted Radon transforms. We also suggest formulas for finding these constants for the general case.