Positive-definite kernel

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A complex-valued function on , where is any set, which satisfies the condition for any , ,  . The measurable positive-definite kernels on a measure space correspond to the positive integral operators (cf. Integral operator) on ; in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces .
The theory of positive-definite kernels extends the theory of positive-definite functions (cf. Positive-definite function) on groups: For a function on a group to be positive definite it is necessary and sufficient that the function on is a positive-definite kernel. In particular, certain results from the theory of positive-definite functions can be extended to positive-definite kernels. For example, Bochner's theorem is that each positive-definite function is the Fourier transform of a positive bounded measure (i.e. an integral linear combination of characters), and this is generalized as follows: Each (generalized) positive-definite kernel has an integral representation by means of so-called elementary positive-definite kernels with respect to a given differential expression .