Examples
For example, when X is finite and H consists of all complex-valued functions on X, then an element of H can be represented as an array of complex numbers. If the usual inner product is used, then Kx is the function whose value is 1 at x and 0 everywhere else, and K(x,y) can be thought of as an identity matrix since K(x,y)=1 when x=y and K(x,y)=0 otherwise. In this case, H is isomorphic to .
A more sophisticated example is the Hardy space H2(D), the space of square-integrable holomorphic functions on the unit disc. So here X=D, the unit disc. It can be shown that the reproducing kernel for H2(D) is
This kernel is an example of a Bergman kernel, named for Stefan Bergman.
Read more about this topic: Reproducing Kernel Hilbert Space
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