Strictly positive definite functions on a real inner product space

If f(t)=Sigma(k=0)(infinity) a(k)t(k) converges for all tis an element ofR with all coefficients a(k)greater than or equal to0, then the function f (<x,y>) is positive definite on H x H for any inner product space H. Set K={k:a(k)>0}. We show that f (<x,y>) 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.