# Is $cos^n(x^2-y^2)$ a valid mercer kernel function?

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 $$\phi(x) = (cos(x^2), sin(x^2))^T$$

• What's your definition of a valid kernel? Does this satisfy that definition?
– Sycorax
Apr 7, 2020 at 22:22
• $\cos(.)$ can be negative, is this an issue? Apr 8, 2020 at 7:51
• kernel function is explained here: stats.stackexchange.com/questions/152897/… Apr 8, 2020 at 9:32
• 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.
– Sycorax
Apr 8, 2020 at 12:48
• 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?
– Sycorax
Apr 8, 2020 at 14:13