The direct and inverse problem for a class of entire functions related to sine type functions

Abstract

For a > 0 and nonnegative integers m and n the classes Aa m,n of entire functions are introduced. An entire function ω belongs to Aa m,n if ω is real on the imaginary axis and has the form (iμ)n−m+1ω(μ) = χ1(μ) cosμa + iχ2(μ) sin μa + (μ), where χ1 is a polynomial of degree n + 1 with leading coefficient i n+1, χ2 is a polynomial of degree at most n, and is small with respect to the other terms. When m = 0, these functions are sine type functions. The functions in Aa m,n have infinitely many zeros with explicitly given asymptotic behaviour, for which the first n + 1 terms in the asymptotics are determined by the coefficients in the polynomials χ1 and χ2. Conversely, any sequence of complex numberswith such an asymptotic representation determines a unique function from a class Aa m,n whose zeros are this given sequence. Some explicit formulas for the direct and the inverse problem are provided.