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发表于 2009-12-14 02:06
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ＲＫＨＳ 中文翻译叫什么？我看英文实在头大，想找本中文书看看
x-valued functions on X. We say that H is a reproducing kernel Hilbert space if every linear map of the form
L_{x} : f \mapsto f(x)
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:
f(x) = \langle f,\ K_x \rangle \quad \forall f \in H \quad (*).
The function Kx is called the point-evaluation functional 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 K: X \times X \to \mathbb{C} by
K(x,y) \ \stackrel{\mathrm{def}}{=}\ \overline{K_x(y)}.
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.
[edit] 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 \mathbb{C}^n.
A more sophisticated example is the Hardy space H2(D), the space of squareintegrable 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
K(x,y)=\frac{1}{\pi}\frac{1}{(1-x\overline{y})^2}.
This kernel is an example of a Bergman kernel, named for Stefan Bergman.
[edit] Properties
[edit] The reproducing property
It is clear from the discussion above that
K(x,y) \;=\; \overline{K_x(y)} \;=\; \langle K_y,K_x\rangle.
In particular,
K(x,x) \;=\; \langle K_x, K_x \rangle \;\geq\; 0, \quad \forall x\in X.
Note that
K_x \;=\; 0 \quad \text{ if and only if } \quad f(x) = 0 \quad \forall \; f\in H.
[edit] Orthonormal sequences
If \textstyle \left\{ \phi_{k}\right\} _{k=1}^{\infty} is an orthonormal sequence such that the closure of its span is equal to H, then
K\left( x,y\right) =\sum_{k=1}^{\infty}\phi_{k}\left( x\right) \overline{\phi _{k}\left( y\right)}. |
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发表于 2009-12-14 02:02
