nLab reproducing kernel Hilbert space

Idea

A reproducing kernel Hilbert space is a Hilbert space of functions in which point evaluation is a continuous linear functional. Using spectral measures one makes connection to specific kind of integral kernels.

Related pages

• projection measure, Bargmann-Segal transform, coherent state, Fock-Bargmann space?
• Bergman kernel

Using reproducing kernels in the context of machine learning is known as the kernel method.

In probability theory, the analogues are Markov kernels (related to Chapman-Kolmogorov formula? for conditional probabilities), see monads of probability, measures, and valuations.

The analogue for quantum amplitudes is used in derivation of Feynman integral approach, including the formal coherent state path integrals.

Literature

• wikipedia: reproducing kernel Hilbert space, Bergman kernel
• N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337–404
• Valentine Bargmann, On a Hilbert space of analytic functions and an associated integral transform, Communications on Pure and Applied Mathematics 14 (1961) 187–214 MR0157250 doi
• J. H. Rawnsley, Coherent states and Kähler manifolds, Quart. J. Math. Oxford (2), 28 (1977) 403–415
• V. V. Kisil, Integral representations and coherent states, Bulletin of the Belgian Mathematical Society, v. 2 (1995), No 5, pp. 529-540.
• Daniel Beltiţă, José E. Galé, Universal objects in categories of reproducing kernels, Rev. Mat. Iberoamericana 27:1 (2011) 123–179 arXiv:0912.0091 MR2815734 euclid
• Daniel Beltiţă, Tudor S. Ratiu, Geometric representation theory for unitary groups of operator algebras, Advances in Mathematics 208:1 (2007) 299–317 doi arXiv:math.RT/0501057