Strictly Hermitian Positive Definite Functions

Let H be any complex inner product space with inner product <, >. We say that f : C -->C is Hermitian positive definite on H if the matrix $$(f(<z^r,z^s>){)}_{r,s=1}^n\text{eqno}(\ast )$$ is Hermitian positive definite for all choice of z^1,…,z^n in H, all n. It is strictly Hermitian positive definite if the matrix (*) is also non-singular for any choice of distinct z^1,…,z^n in H. In this article we prove that if dim H >= 3, then f is Hermitian positive definite on H if and only if $$f(z)=\sum _{k,m=0}^{\mathrm{\infty }}{b}_{k,m}{z}^k{\text{oz}}^m\text{eqno}(\ast \ast )$$ where \oz is the conjugate of z, b_{k,m}>= 0, all k,m in Z_+, and the series converges for all z in C. We also prove that f of the form (**) is strictly Hermitian positive definite on any H if and only if the set $$J=(k,m):b_{k,m}>0$$ is such that (0,0) is in J, and every arithmetic sequence in Z intersects the values {k-m : (k,m)\in J} an infinite number of times.