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\cos(x) = \frac{1}{2} ( e^{i x} + e^{- i x}) = \frac{1}{2}(g_{1} + g_{-1}).
</math>
One can create a positive definite function <math>f \colon X \to \mathbb{C}</math> easily from positive definite function <math>f \colon \R \to \mathbb C</math> for any [[vector space]] <math>X</math>: choose a [[linear function]] <math>\phi \colon X \to \R</math> and define <math>f^* := f \circ \phi</math>.
Then
:<math>
u^* A^{(f^*)} u
= \sum_{j, k = 1}^{n} \overline{u_k} u_j f^*(x_k - x_j)
= \sum_{j, k = 1}^{n} \overline{u_k} u_j f(\phi(x_k) - \phi(x_j))
= u^* \tilde{A}^{(f)} u
\ge 0,
</math>
where <math>\tilde{A}^{(f)} = \big( f(\phi(x_i) - \phi(x_j)) = f(\tilde{x}_i - \tilde{x}_j) \big)_{i, j}</math> where <math>\tilde{x}_k := \phi(x_k)</math> are distinct as <math>\phi</math> is [[linear]].
===Bochner's theorem===
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