Difference between revisions of "Positive-definite function on a group"
From Encyclopedia of Mathematics
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| − | A continuous [[Function|function]] | + | {{TEX|done}} |
| + | A continuous [[Function|function]] $f\neq0$ on the [[Group|group]] $G$ such that for all $x_1,\dots,x_n$ in $G$ and $c_1,\dots,c_n\in\mathbf C$, | ||
| − | + | $$\sum_{i,j}f(x_ix_j^{-1})c_i\overline{c_j}\geq0.$$ | |
| − | Examples can be obtained as follows. Let | + | Examples can be obtained as follows. Let $\pi\colon G\to\Aut(H)$ be a [[Unitary representation|unitary representation]] of $G$ in a [[Hilbert space|Hilbert space]] $H$, and let $u$ be a unit (length) vector. Then |
| − | + | $$f(x)=\langle\pi(x)u,u\rangle$$ | |
is a positive-definite function. | is a positive-definite function. | ||
| − | Essentially, these are the only examples. Indeed, there is a [[Bijection|bijection]] between positive-definite functions on | + | Essentially, these are the only examples. Indeed, there is a [[Bijection|bijection]] between positive-definite functions on $G$ and isomorphism classes of triples $(\pi,H,u)$ consisting of a unitary representation $\pi$ of $G$ on $H$ and a unit vector $u$ that topologically generates $H$ under $\pi(G)$ (a [[Cyclic vector|cyclic vector]]). This is the (generalized) Bochner–Herglotz theorem. |
| − | See also [[Fourier–Stieltjes transform|Fourier–Stieltjes transform]] (when | + | See also [[Fourier–Stieltjes transform|Fourier–Stieltjes transform]] (when $G=\mathbf R$). |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lang, " | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Lang, "$\SL_2(\mathbf R)$" , Addison-Wesley (1975) pp. Chap. IV, §5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> G.W. Mackey, "Unitary group representations in physics, probability and number theory" , Benjamin (1978) pp. 147ff</TD></TR></table> |
Latest revision as of 17:51, 28 November 2018
A continuous function $f\neq0$ on the group $G$ such that for all $x_1,\dots,x_n$ in $G$ and $c_1,\dots,c_n\in\mathbf C$,
$$\sum_{i,j}f(x_ix_j^{-1})c_i\overline{c_j}\geq0.$$
Examples can be obtained as follows. Let $\pi\colon G\to\Aut(H)$ be a unitary representation of $G$ in a Hilbert space $H$, and let $u$ be a unit (length) vector. Then
$$f(x)=\langle\pi(x)u,u\rangle$$
is a positive-definite function.
Essentially, these are the only examples. Indeed, there is a bijection between positive-definite functions on $G$ and isomorphism classes of triples $(\pi,H,u)$ consisting of a unitary representation $\pi$ of $G$ on $H$ and a unit vector $u$ that topologically generates $H$ under $\pi(G)$ (a cyclic vector). This is the (generalized) Bochner–Herglotz theorem.
See also Fourier–Stieltjes transform (when $G=\mathbf R$).
References
| [a1] | S. Lang, "$\SL_2(\mathbf R)$" , 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