×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: EuDML
References:
[1] EDWARDS D. A.: On the continuity properties of functions satisfying a condition of Sirvint’s. Quart. J. Math. Oxford 8, 1957, 58-67. · Zbl 0077.31203
[2] EDWARDS R. E.: Functional Analysis. Holt, Rinehart and Winston, New York 1965. · Zbl 0182.16101
[3] EDWARDS R. E.: Fourier Series, vol. 2. Springer-Verlag, Berlin 1982. · Zbl 0599.42001
[4] HILLE E., PHILLIPS R. S.: Functional Analysis and Semigroups. AMS Providence, New York 1957. · Zbl 0078.10004
[5] MARINESCU G.: Espaces Vectoriels Pseudotopologiques et Théorie des Distributions. Akademie-Verlag, Berlin 1963. · Zbl 0124.31602
[6] SIRVINT G.: Weak compactness in Banach spaces. Studia Math. 11, 1950, 71-94. · Zbl 0045.37903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.