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How to show if $cos^n(x^2-y^2)$ is a valid mercer kernel function if $n$ is positive?

For $cos(x^2-y^2)$ I would assume that: $cos(x^2-y^2) = sin(x^2)sin(y^2)+cos(x^2)cos(y^2)$ Is a valid mercer kernel with feature map $\phi(x) = (cos(x^2), sin(x^2))^T$

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    $\begingroup$ What's your definition of a valid kernel? Does this satisfy that definition? $\endgroup$
    – Sycorax ♦
    Apr 7, 2020 at 22:22
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    $\begingroup$ $\cos(.)$ can be negative, is this an issue? $\endgroup$
    – Xi'an
    Apr 8, 2020 at 7:51
  • $\begingroup$ kernel function is explained here: stats.stackexchange.com/questions/152897/… $\endgroup$
    – user1727556
    Apr 8, 2020 at 9:32
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    $\begingroup$ I'm not asking what a kernel is abstractly. I'm asking what your definition is. The word "kernel" is used differently in different places, and knowing what definition you use is important. $\endgroup$
    – Sycorax ♦
    Apr 8, 2020 at 12:48
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    $\begingroup$ The question you've linked asks for intuitive explanations about what a kernel is; none of the answers provide a definition. For example, a Mercer kernel is (1) symmetric and (2) positive semidefinite. Neither of these criteria is expressly manifest on the linked page. Are you asking about Mercer kernels? Or something else? $\endgroup$
    – Sycorax ♦
    Apr 8, 2020 at 14:13

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