Reproducing kernel Hilbert space: Difference between revisions

Revision as of 14:20, 11 August 2006

Revision as of 14:26, 11 August 2006

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== Moore-Aronszajn theorem ==

Given a positive definite kernel <math>K</math>, we can construct a unique RKHS <math>H</math> with <math>K</math> as the reproducing kernel. A positive definite kernel is a function on <math>TX \otimes TX</math> with the following property.

For all natural number <math>n</math>, for all <math>t_1x_1, \ldots, t_nx_n</math> in <math>TX</math>, and for all <math>\alpha_1, \ldots, \alpha_n</math> in a real or complex,

:<math> \sum_{i=1}^n \sum_{j=1}^n \alpha_i \alpha_j K(t_ix_i, t_jx_j) \ge 0.</math>

Now, for all <math>tx</math> in <math>TX</math> define the following functions as the first class citizens of <math>H</math>

:<math> \Phif(tx) = K(\cdot, tx). </math>

Let <math>H'</math> be the linear [[vector space]] [[linear span|spanned]] by the set <math>\{ \Phif(tx) \}_{tx \in TX}</math>.

Finally we complete <math>H'</math> by including all the [[Cauchy sequence]]s of <math>H'</math>.