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On Herglotz theorem in vector lattices. (English) Zbl 0876.42004
It is well-known that it is possible to characterize Fourier-Stieltjes coefficients of the (right-continuous) non-decreasing, bounded functions as positive definite sequences. Recall that a numerical sequence \((a_n)^\infty_{n=-\infty}\) is said to be positive definite if for any (complex) sequence \((z_n)\) having only a finite number of terms different from zero we have \(\sum_{n,m} a_{n-m}z_n\overline z_m\geq 0\).
Now according to the Herglotz theorem a numerical sequence \((a_n)^\infty_{n=-\infty}\) is positive definite if and only if there exists a right-continuous, non-decreasing, bounded function \(F\) on \([-\pi,\pi]\) with \(F(-\pi)=0\), such that \(a_n=\int_{(-\pi,\pi]} e^{-ins}dF(s)\) for all \(n=0,\pm1,\dots\) .
In this paper, we give a generalization of the Herglotz theorem for \(a_n\) being elements of a vector lattice.

42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.)