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
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$$(f(< z^r ,z^s ))_{r,s = 1}^n $$
is Hermitian positive definite for all choice of*z*^{1},…,*z*^{n} in*H* for 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
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$$f(z) = \sum\limits_{k,\ell = 0}^\infty {b_{k.\ell ^{z^k z^{ - \ell } ,} } } $$
where*b*_{k,l}≥0 for all*k, l* 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,l*):*b*_{k,l}>0} is such that (0,0)∈*J*, and every arithmetic sequence in ℤ intersects the values {*k*−*l*: (*k, l*)∈*J*} an infinite number of times.