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