Generalized herglotz domains

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Abstract
Let F(θ k, α) be the far field pattern arising from the scattering of a time harmonic plane acoustic wave of wave number k and direction a by a sound‐soft cylinder of cross section D. Suppose F has the Fourier expansion magnified image where an = an(k, . Then if ϰ2 is a Dirichlet eigenvalue for D, sufficient conditions are given on D for the existence of a nontrivial sequence |bn| where the bn are independent of such that for all directions magnified image Domains 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 ϱD or in terms of the Rayleigh hypothesis in acoustic scattering theory and examples are given showing the applicability of these conditions.