Generalized herglotz domains

摘要

AbstractLetF(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave numberkand directionaby a sound‐soft cylinder of cross sectionD. SupposeFhas the Fourier expansionwherean=an(k, . Then if ϰ2is a Dirichlet eigenvalue forD, sufficient conditions are given onDfor the existence of a nontrivial sequence bn where thebnare independent of such that for all directionsDomains for which this is true are called generalized Herglotz domains. The conditions for a domain to be a generalized Herglotz domain are given either in terms of the Schwarz function for the analytic boundary ϱDor in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these condi