Does there exist a (stationary) covariance kernel function which is $C^\infty$-smooth but not real analytic? If so, could you please provide an example?
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2 Answers
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With $H=\mathbf 1_{\mathbb R_+}$, $t,x$ real, $ H(t)t^{-1/2}e^{-x^2/t} $ is the fundamental solution of the heat equation, $C^\infty$ everywhere except at $(0,0)$, analytic only outside $t=0$.
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Take a schwarz class function f with compact support. It won't be real analytic for that reason at least.$g = f(x)*f(-x)$ is the fourier transform of the non-negative function $|\hat f|^2$, and it is smooth with compact support. By Bochner's thm it is a co-variance kernel.
