Positive-definite function on a group

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A continuous function on the group such that for all in and ,

Examples can be obtained as follows. Let be a unitary representation of in a Hilbert space , and let be a unit (length) vector. Then

is a positive-definite function.

Essentially, these are the only examples. Indeed, there is a bijection between positive-definite functions on and isomorphism classes of triples consisting of a unitary representation of on and a unit vector that topologically generates under (a cyclic vector). This is the (generalized) Bochner–Herglotz theorem.

See also Fourier–Stieltjes transform (when ).


[a1] S. Lang, "" , Addison-Wesley (1975) pp. Chap. IV, §5
[a2] G.W. Mackey, "Unitary group representations in physics, probability and number theory" , Benjamin (1978) pp. 147ff
How to Cite This Entry:
Positive-definite function on a group. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article