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In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space associated with a kernel that reproduces every function in the space or, equivalently, where every evaluation functional is bounded. The reproducing kernel was first introduced in the 1907 work of Stanisław Zaremba concerning boundary value problems for harmonic and biharmonic functions. James Mercer simultaneously examined functions which satisfy the reproducing property in the theory of integral equations. The idea of the reproducing kernel remained untouched for nearly twenty years until it appeared in the dissertations of Gábor Szegő, Stefan Bergman, and Salomon Bochner. The subject was eventually systematically developed in the early 1950s by Nachman Aronszajn and Stefan Bergman. 〔Okutmustur〕 These spaces have wide applications, including complex analysis, harmonic analysis, and quantum mechanics. Reproducing kernel Hilbert spaces are particularly important in the field of statistical learning theory because of the celebrated Representer theorem which states that every function in an RKHS can be written as a linear combination of the kernel function evaluated at the training points. This is a practically useful result as it effectively simplifies the empirical risk minimization problem from an infinite dimensional to a finite dimensional optimization problem. For ease of understanding, we provide the framework for real-valued Hilbert spaces. The theory can be easily extended to spaces of complex-valued functions and hence include the many important examples of reproducing kernel Hilbert spaces that are spaces of analytic functions. 〔Paulson〕 ==Definition== Let ''X'' be an arbitrary set and ''H'' a Hilbert space of real-valued functions on ''X''. The evaluation functional over the Hilbert space of functions ''H'' is a linear functional that evaluates each function at a point ''x'', : We say that ''H'' is a reproducing kernel Hilbert space if is a continuous function for any in or, equivalently, if for all in ''X'', is a bounded operator on , i.e. there exists some ''M > 0'' such that While property () is the weakest condition that ensures both the existence of an inner product and the evaluation of every function in ''H'' at every point in the domain, it does not lend itself to easy application in practice. A more intuitive definition of the RKHS can be obtained by observing that this property guarantees that the evaluation functional can be represented by taking the inner product of with a function in ''H'' . This function is the so-called reproducing kernel for the Hilbert space ''H'' from which the RKHS takes its name. More formally, the Riesz representation theorem implies that for all ''x'' in ''X'' there exists a unique element of ''H'' with the reproducing property, Since is itself a function in ''H'' we have that for each ''y'' in ''X'' : This allows us to define the reproducing kernel of ''H'' as a function by : From this definition it is easy to see that a function is a reproducing kernel if it is both symmetric and positive definite, i.e. : for any 〔 Durrett 〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Reproducing kernel Hilbert space」の詳細全文を読む スポンサード リンク
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