# zbMATH — the first resource for mathematics

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.

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