# Positive-definite function on a group

From Encyclopedia of Mathematics

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 ).

#### References

[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: http://encyclopediaofmath.org/index.php?title=Positive-definite_function_on_a_group&oldid=18615

This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article