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On extension of scalar valued positive definite functions on ordered groups. (English) Zbl 1152.43003

It was proven by M. G. Krein in 1940 that every continuous positive definite function on \(I=\left(-a,a\right)\) can be extended to a continuous positive definite function on the real line. It is natural to expect that every continuous positive definite function, on a symmetric neighborhood of the neutral element of a locally compact abelian group, has a positive definite extension to the whole group. However, this statement is not true [see, e.g., A. Devinatz, Acta Math. 102, 109–134 (1959; Zbl 0100.10601)].
The authors deal with the notion of symmetric generalized interval and prove that the extension is possible even without the requirement of continuity of a function.
Definition. Let \(\Gamma\) be an ordered group. A nonempty set \(J\) contained in \(\Gamma\) is a generalized interval if \(J\) has the following property: \(a,b\in J\), \(a<b\) implies \(\left(a,b\right)\subset J\). We say that \(J\) is not trivial if \(J\) has more than one point.
The main result of the article is the following
Theorem. Let \(\left(\Gamma,+\right)\) be an abelian ordered group and let \(\Delta\subset\Gamma\) be a nontrivial symmetric generalized interval.
Let \(f:\,\Delta\to{\mathbb C}\) be a positive definite function. Then: 7mm
(a)
\(f\) has a positive definite extension to the whole group \(\Gamma\).
(b)
If \(\Gamma\) is a topological group and \(f\) is continuous then any positive definite extension of \(f\) is continuous.
(c)
If \(\Gamma\) is a locally compact group and \(f\) is measurable then any positive definite extension of \(f\) is measurable.
(d)
If \(\Gamma\) is a locally compact group and \(f\) is measurable then there exist two positive definite functions \(f^c:\,\Delta\to{\mathbb C}\) and \(f^0:\,\Delta\to{\mathbb C}\) such that
8mm
(i)
\(f=f^c+f^0\).
(ii)
\(f^c\) is continuous.
(iii)
\(f^0\) is zero locally almost everywhere.
The article also contains a history of the problem and a bibliography that reflects several results obtained in this area. It should be very interesting for specialists in harmonic analysis, approximation theory and other areas of mathematics.

MSC:

43A35 Positive definite functions on groups, semigroups, etc.
42A82 Positive definite functions in one variable harmonic analysis

Citations:

Zbl 0100.10601
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