Reproducing kernel Hilbert space: Difference between revisions

Revision as of 11:57, 21 October 2024

Revision as of 19:30, 31 October 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 ifconvergence twoin functionsnorm implies <math>f</math>pointwise convergence (and it <math>g</math>implies uniform in the RKHS are close in norm (i.e.,convergence if <math> \|fsup_{x - g\|</math>in X} is small), then <math>f</math> and <math>g</math> are also pointwise close (i.e., <math>\sup|| K_x||f(x) - g(x)|</math> is smallfinite). This follows from the fact that the inner product induces pointwise evaluation control. Roughly speakingHowever, this means that closeness in the RKHS norm implies pointwise closeness, but 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]].