Einar Hille

Copyright © 1980 Einar Hille. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let f(z) be holomorphic in the strip −σ<y<σ<∞ and satisfy the conditions for having an expansion in an Hermitian series f(z)=∑n=0∞fnhn(z),   hn(z)=(π122nn!)−12e−12z2Hn(z),absolutely convergent in the strip. Two meanvalues 𝔐k(f;y)={π−12∫−∞∞e−kx2|f(x+iy)|2dz}12,   k=0,1.are discussed, directly using the condition on f(z) or via the Hermitian series. Integrals involving products hm(x+iy)hn(x−iy) are discussed. They lead to expansions of the mean squared in terms of Laguerre functions of y2 when k=0 and in terms of Hermite functions hn(212iy) when k=1. The sumfunctions are holomorphic in y. They are strictly increasing when |y| increases.