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On vector measures and distributions. (English) Zbl 0684.46038
Let X be a sequentially complete locally convex Hausdorff space and let C be the space of all continuous, complex-valued functions on \({\mathbb{R}}\) with period \(2\pi\). A Radon measure is a weakly compact linear mapping F from C into X. The author considers the series:
(i) \(\sum_{n}c_ ne^{int}\), \(c_ n\in X\) and the formally integrated series;
(ii) \(\sum_{n\neq 0}(in)^{-1}c_ ne^{int}.\)
It is shown that (i) is the Fourier-Stieltjes series of a Radon measure \((<F,e^{int}>=c_ n)\) if and only if (ii) is the Fourier-Lebesgue series of some function z: [-\(\pi\),\(\pi\) ]\(\to X\) with weakly compact semi-variation. It is deduced from this result that any Radon measure F has the form \(F=c+Dz\), where \(c\in X\) and Dz is the distributional derivative of z.
Reviewer: Ch.Swartz
46G10 Vector-valued measures and integration
46F99 Distributions, generalized functions, distribution spaces
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