On the transfer of information in multiplier equations

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.