Reproducing kernel Hilbert space: Difference between revisions

Revision as of 18:58, 25 February 2006

Revision as of 22:39, 25 February 2006

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:<math> f \mapsto f(x) </math>

from ''H'' to the complex numbers is continuous for any ''x'' in ''X''. By the [[Riesz representation theorem]], this implies that for given ''x'' there exists an element ''K''_''x'' of ''H'' suchwith thatthe forproperty every function ''f'' in the space,that:

:<math> f(x) = \langle K_x, f \rangle. \quad \forall f \in H \quad (*)</math>

The function

:<math> K(x,y) =\equiv K_x(y) </math>

is called a reproducing kernel for the Hilbert space. In fact, ''K'' is uniquely determined by the above condition (*). In some concrete contexts this amounts to saying

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 ''K''_''x'' is the function whose value is 1 at ''x'' and 0 everywhere else.

:<math>f(x)=\int_\Omega K(x,y) f(y)\,dy</math>

In other contexts, (*) amounts to saying

for every ''f'', where &Omega; is the appropriate ___domain, often the real numbers or '''R'''^''n''.

:<math>f(x)=\int_\Omegaint_X K(x,y) f(y)\,dy</math>

for every ''f'', where &Omega;''X'' is the appropriate ___domain, often the real numbers or '''R'''^''n''.

==Bergman kernel==