Definition
Let X be an arbitrary set and H a Hilbert space of complex-valued functions on X. We say that H is a reproducing kernel Hilbert space if the linear map
from H to the complex numbers is continuous for any x in X. By the Riesz representation theorem, this implies that for every x in X there exists a unique element Kx of H with the property that:
The function Kx is called the point-evaluation function at the point x.
Since H is a space of functions, the element Kx is itself a function and can therefore be evaluated at every point. We define the function by
This function is called the reproducing kernel for the Hilbert space H and it is determined entirely by H because the Riesz representation theorem guarantees, for every x in X, that the element Kx satisfying (*) is unique.
Read more about this topic: Reproducing Kernel Hilbert Space
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