Positive-definite function on a group
From Encyclopedia of Mathematics
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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 ).
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
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