If f(t) = ∑_(k=0)~∞a_kt~k converges for all t ∈ R with all coefficients a_k ≥ 0, then the function f() is positive definite on H * H for any inner product space H. Set K = {k: a_k > 0}. We show that f() is strictly positive definite if and only if K contains the index 0 plus an infinite number of even integers and an infinite number of odd integers.
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