Contributions to the theory of Hermitian series. II. The representation problem

L Introduction.In the first note of this series [2]f the author laid a broad foundation for the theory of Hermitian series in the complex domain.A number of questions encountered during this investigation were merely mentioned and detailed discussion had to be postponed till later communications.The representation or expansion problem was such a question, that is, the problem of finding necessary and sufficient conditions in order that an analytic function shall be representable by an Hermitian series for complex values of the variable.This problem is solved in the present note.Let ff»(z) denote the wth normalized orthogonal function of Hermite dn (1.1) H"(z) = [tt1'^"»!]-1^-1)v2'2-■[e-°2].dzn Iif(x) is a measurable function such that xn exp [ -x3/2] f(x) e L\( -», «o) for n = 0, 1, 2, • • • , then/(x) has an associated Fourier-Hermite series (1.2) /(*)~ £/«»,(*) n=0 where /. = I f(t)Hn{t)dt.J -oo This series is ordinarily not convergent, but if we assume for instance that exp [ -ax2]f(x) t Li(-w, °o) for somea