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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
##### MSC:
 46G10 Vector-valued measures and integration 46F99 Distributions, generalized functions, distribution spaces
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##### References:
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