# Positive-definite function

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A complex-valued function $\phi$ on a group $G$ satisfying

$$\sum _ {i,j= 1 } ^ { m } \alpha _ {i} \overline \alpha \; _ {j} \phi ( x _ {j} ^ {-} 1 x _ {i} ) \geq 0$$

for all choices $x _ {1} \dots x _ {m} \in G$, $\alpha _ {1} \dots \alpha _ {m} \in \mathbf C$. The set of positive-definite functions on $G$ forms a cone in the space $M( G)$ of all bounded functions on $G$ which is closed with respect to the operations of multiplication and complex conjugation.

The reason for distinguishing this class of functions is that positive-definite functions define positive functionals (cf. Positive functional) on the group algebra $\mathbf C G$ and unitary representations of the group $G$( cf. Unitary representation). More precisely, let $\phi : G \rightarrow \mathbf C$ be any function and let $l _ \phi : \mathbf C G\rightarrow \mathbf C$ be the functional given by

$$l _ \phi \left ( \sum _ {g \in G } \alpha _ {g} g \right ) = \ \sum _ {g \in G } \phi ( g) \alpha _ {g} ;$$

then for $l _ \phi$ to be positive it is necessary and sufficient that $\phi$ be a positive-definite function. Further, $l _ \phi$ defines a $*$- representation of the algebra $\mathbf C G$ on a Hilbert space $H _ \phi$, and therefore a unitary representation $\pi _ \phi$ of the group $G$, where $\phi ( g) = ( \pi _ \phi ( g) \xi , \xi )$ for some $\xi \in H _ \phi$. Conversely, for any representation $\pi$ and any vector $\xi \in H _ \phi$, the function $g \rightarrow ( \pi ( g) \xi , \xi )$ is a positive-definite function.

If $G$ is a topological group, the representation $\pi _ \phi$ is weakly continuous if and only if the positive-definite function is continuous. If $G$ is locally compact, continuous positive-definite functions are in one-to-one correspondence with the positive functionals on $L _ {1} ( G)$.

For commutative locally compact groups, the class of continuous positive-definite functions coincides with the class of Fourier transforms of finite positive regular Borel measures on the dual group. There is an analogue of this assertion for compact groups: A continuous function $\phi$ on a compact group $G$ is a positive-definite function if and only if its Fourier transform $\widehat \phi ( b)$ takes positive (operator) values on each element of the dual object, i.e.

$$\int\limits _ { G } \phi ( g) ( \sigma ( g) \xi , \xi ) dg \geq 0$$

for any representation $\sigma$ and any vector $\xi \in H _ \sigma$, where $H _ \sigma$ is the space of $\sigma$.

How to Cite This Entry:
Positive-definite function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive-definite_function&oldid=48249
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article