An automorphism of the space of analytic functions, having expansions in Hermite polynomials

Abstract

For each fixed τ0 > 0 we denote by S(τ0) = {z∈C:|IMz|<τ0} a strip parallel to the real axes. Further by H(τ0) we denote the Frèshet space of all analytic functions, defined in S(τ0) and having representation there in series of Hermite polynomials. Let α be a complex number such that −1/2 < Re α < 1/2. We consider the mapping P(α):H(τ0)→H(τ0), (1) defined by the equality f(z) = P(α)(F)(z) = 1Γ(α+1/2) ∫ 01(1−t2)α−1/2F(zt)dt,F∈H(τ0), (2) where Γ(.) is the Gamma-function. We prove that the operator P(α) is an automorphism of the space H(τ0).