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
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
—François, Duc De La Rochefoucauld (1613–1680)
“The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
—Ralph Waldo Emerson (1803–1882)