# zbMATH — the first resource for mathematics

On Herglotz theorem in partially ordered vector spaces. (English) Zbl 0994.42002
In the paper the Herglotz theorem is generalized for partially ordered spaces. Let $$Y$$ denote a real monotone $$\sigma$$-complete partially ordered vector space and $$Z$$ denote the complexification of $$Y$$. For a sequence $$z_j$$ of elements of $$Z$$ let $$\sigma _N(z_j,t)$$ be the sequence of Cesàro sums of the sequence $$z_j$$. If for all real $$t$$ and natural $$N$$, $$\sigma _N(z_j,t) \geq 0$$, then there exists a positive linear mapping $$L\:C_{2\pi }(\mathbb R)\to Y$$ for which the sequence $$z_j$$ is the sequence of Fourier coefficients, $$C_{2\pi }(\mathbb R)$$ being the space of the continuous $$2\pi$$-periodic functions on the reals $$\mathbb R$$.
##### 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.)
Full Text: