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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 \(\mathbb R\).
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.)
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