Positive-definite kernel

A complex-valued function $ K $ on $ X \times X $, where $ X $ is any set, which satisfies the condition

$$ \sum _ {i,j= 1 } ^ { n } K( x _ {i} , x _ {j} ) \lambda _ {i} \overline \lambda \; _ {j} \geq 0, $$

for any $ n \in \mathbf N $, $ \lambda _ {i} \in \mathbf C $, $ x _ {i} \in X $ $ ( i = 1 \dots n) $. The measurable positive-definite kernels on a measure space $ ( X, \mu ) $ correspond to the positive integral operators (cf. Integral operator) on $ L _ {2} ( X, \mu ) $; in order to include arbitrary positive operators in this correspondence one has to introduce generalized positive-definite kernels, which are associated with Hilbert spaces [1].

The theory of positive-definite kernels extends the theory of positive-definite functions (cf. Positive-definite function) on groups: For a function $ f $ on a group $ G $ to be positive definite it is necessary and sufficient that the function $ K( x, y) = f( xy ^ {-} 1 ) $ on $ G \times G $ 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 [1].

References

[1]	Yu.M. [Yu.M. Berezanskii] Berezanskiy, "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc. (1968) (Translated from Russian)
[2a]	M.G. Krein, "Hermitian positive kernels on homogeneous spaces I" Ukr. Mat. Zh. , 1 : 4 (1949) pp. 64–98 (In Russian)
[2b]	M.G. Krein, "Hermitian positive kernels on homogeneous spaces II" Ukr. Mat. Zh. , 2 : 1 (1950) pp. 10–59 (In Russian)

References

[a1]	H. Reiter, "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press (1968)