Strictly Positive Definite Functions on a Real Inner Product Space

Abstract

If $f(t) = \sum\nolimits_{k = 0}^\infty {a_k t^k }$ converges for all $.......CONVERSION........$ with all coefficients $a_k \geqslant {\text{0}}$, then the function $f(\left\langle {x,\left. y \right\rangle } \right.)$ is positive definite on H×H for any inner product space H. Set K={k: a _k >0}. We show that $f(\left\langle {{\text{x,}}\left. {\text{y}} \right\rangle } \right.)$ 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.