Hermite series as boundary values

Abstract

The relations between Hermite series expansions of functions and tempered distributions on the real axis and holomorphic or harmonic functions or generalizations of them in the upper half plane are studied. The Hermite series expansions of H 2 {H^2} functions are characterized in terms of their coefficients. Series of analytic representations of Hermite functions, series of Hermite functions of the second kind, and combined series of Hermite functions of the first and second kind are investigated. The functions to which these series converge in the upper half plane are shown to approach (in various ways) the distributions or functions whose Hermite series have the same coefficients.