Reproducing kernel Hilbert space: Difference between revisions

Revision as of 18:52, 25 February 2006

Revision as of 18:58, 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'' withsuch thethat propertyfor that:every function ''f'' in the space,

:<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

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

When ''X'' is finite, then a complex-valued function on ''X'' 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.

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

In other contexts, (*) amounts to saying

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

for every ''f'', where ''X'' is often the real numbers or '''R'''^''n''.

==Bergman kernel==