In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space is a Hilbert space of functions in which pointwise evaluation is a continuous linear functional. Equivalently, they are spaces that can be defined by reproducing kernels. The subject was originally and simultaneously developed by Nachman Aronszajn (1907–1980) and Stefan Bergman (1895–1977) in 1950.
In this article we assume that Hilbert spaces are complex. The main reason for this is that many of the examples of reproducing kernel Hilbert spaces are spaces of analytic functions, although some real Hilbert spaces also have reproducing kernels.
An important subset of the reproducing kernel Hilbert spaces are the reproducing kernel Hilbert spaces associated to a continuous kernel. These spaces have wide applications, including complex analysis, harmonic analysis, quantum mechanics, statistics and machine learning.
Read more about Reproducing Kernel Hilbert Space: Definition, Examples, Moore-Aronszajn Theorem, Bergman Kernel
Famous quotes containing the words kernel and/or space:
“We should never stand upon ceremony with sincerity. We should never cheat and insult and banish one another by our meanness, if there were present the kernel of worth and friendliness. We should not meet thus in haste.”
—Henry David Thoreau (1817–1862)
“Here were poor streets where faded gentility essayed with scanty space and shipwrecked means to make its last feeble stand, but tax-gatherer and creditor came there as elsewhere, and the poverty that yet faintly struggled was hardly less squalid and manifest than that which had long ago submitted and given up the game.”
—Charles Dickens (1812–1870)