On the transfer of information in multiplier equations

TL;DR

This paper derives spectral width estimates for solutions of multiplier equations, providing explicit formulas and numerical verification, with applications in inverse source problem stability analysis.

Contribution

It introduces uniform spectral width estimates for tempered solutions of multiplier equations and provides explicit fundamental solutions, verified numerically.

Findings

Spectral width estimates are uniform for solutions up to a certain order.

Explicit expression for a tempered fundamental solution of a multiplier.

Numerical verification confirms the theoretical estimates in various scenarios.

Abstract

We derive spectral width estimates for traces of tempered solutions of a large class of multiplier equations in $\mathbf{R}^n$. The estimates are uniform for solutions up to a given order. In the process, we find a rather explicit expression for a tempered fundamental solution of a multiplier. We successfully verify our spectral width estimates against numerical results in several scenarios involving the inhomogeneous Helmholtz equation in $\mathbf{R}^n$ with $n=1,\dots,9$. Our main result is directly applicable in the stability analysis of solutions of inverse source problems.