Reproducing kernel Hilbert space: Difference between revisions

Revision as of 15:25, 4 December 2024

Revision as of 18:37, 6 December 2024

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The function <math>K_x</math> is called the reproducing kernel, and it reproduces the value of <math>f</math> at <math>x</math> via the inner product.

An immediate consequence of this property is that convergence in norm implies pointwiseuniform convergence (andon itany impliessubset uniformof convergence<math>X</math> ifon which <math> \sup_{x \in X} || K_x\|| </math> is finite)bounded. However, the converse does not necessarily hold.

For example, consider the sequence of functions <math>\sin^{2n}(x)</math>. These functions converge pointwise to 0 as <math>n \to \infty</math> , but they do not converge uniformly (i.e., they do not converge with respect to the supremum norm). This illustrates that pointwise convergence does not imply convergence in norm. It is important to note that the supremum norm does not arise from any inner product, as it does not satisfy the [[Polarization identity|parallelogram law]].