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Change "Most common usage" and "Alternative definition" to "Definition 1" and "Definition 2" |
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In [[mathematics]], a '''positive-definite function''' is, depending on the context, either of two types of [[function (mathematics)|function]].
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A ''positive semi-definite function'' of a [[real number|real]] variable ''x'' is a [[complex number|complex]]-valued function <math> f: \mathbb{R} \to \mathbb{C} </math> such that for any real numbers ''x''<sub>1</sub>, …, ''x''<sub>''n''</sub> the ''n'' × ''n'' [[matrix (mathematics)|matrix]]
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One can define positive-definite functions on any [[locally compact abelian topological group]]; Bochner's theorem extends to this context. Positive-definite functions on groups occur naturally in the [[representation theory]] of groups on [[Hilbert space]]s (i.e. the theory of [[unitary representation]]s).
==Definition 2==
The following definition conflicts with the one above.
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