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===Examples===
If <math>(X, \langle \cdot, \cdot \rangle)</math> is a real [[inner product space]], then <math>g_y \colon X \to \mathbb{C}</math>, <math>x \mapsto \exp(i \langle y, x \rangle)</math> is positive definite for every <math>y \in X</math>: for all <math>u \in \mathbb{C}^n</math> and all <math>x_1, \ldots, x_n</math> we have
:<math>
u^* A^{(g_y)} u
= \sum_{j, k = 1}^{n} \overline{u_k} u_j e^{i \langle y, x_k - x_j \rangle}
= \sum_{k = 1}^{n} \overline{u_k} u_j e^{i \langle y, x_k \rangle} \sum_{j = 1}^{n} u_j e^{- i \langle y, x_j \rangle}
= \left| \sum_{j = 1}^{n} \overline{u_j} e^{i \langle y, x_j \rangle} \right|^2
\ge 0.
</math>
As nonnegative linear combinations of positive definite functions are again positive definite, the [[cosine function]] is positive definite as a nonnegative linear combination:
:<math>
\cos(x) = \frac{1}{2} ( e^{i x} + e^{- i x}) = \frac{1}{2}(g_{1} + g_{-1}).
</math>
===Bochner's theorem===
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