Mercer theorem

The key to verifying a new symmetric function as a kernel is the condition stated in Mercer Theorem, that is, the requirement that the... 

Kernels are symmetric functions. Mercer s theorem provides a characterization of kernels a symmetric function K u, v) is a kernel function if and only if the matrix... 

Theorem 2.1 (Mercer) Let X be a compact subset of R". Suppose K is a continuous symmetric function such that the integral operator T, L,(X) L,(X). 

Let us observe this theorem, the positivity condition I K(x,z)/(x)/(z)dxdz > 0, V/ 6 Z2 (X), corresponds to the positive semi-definite condition in the finite case, this gives the second characterization of a kernel function that is proved to be most useful when we come to constructing kernels. We use the term kernel to refer to function satisfying this property, but in the literature these are often called Mercer kernel. 

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