Contributions to the theory of Hermitian series III. Meanvalues - Einar Hille (International Journal of Mathematics and Mathematical Sciences)
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.