×

Jump functions of a real interval to a Banach space. (English) Zbl 0688.46009

Summary: On the space \(\ell bv(I,X)\) of the functions of a real interval I to a Banach space X, with locally bounded variation, a topology is defined by the semi-norms \(f\mapsto var(f;a,b)\), \(a\in I\), \(b\in I\). By definition, jump functions are the elements of the closure of the space of local step functions. The decomposition of every \(f\in \ell bv(I,X)\) into the sum of a jump function and a continuous element of \(\ell bv(I,X)\) is constructed. Among other properties, it is studied how this decomposition is reflected on the real function \(V_ f\), the indefinite variation of f. The connection of what precedes with the decomposition of a real measure into the sum of a diffuse measure and an atomic measure is investigated.

MSC:

46E27 Spaces of measures
28A33 Spaces of measures, convergence of measures
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] Bourbaki, N.).- Intégration, Hermann (Paris), Chap. 1, 2, 3, 4 (2d ed.), 1965, Chap. 5, 1956; Chap. 6, 1956; Chap. 7,8, 1963.
[2] Dinculeanu, N.).- Vector Measures, Pergamon (London, New-York), 1967. · Zbl 0156.14902
[3] Dunford, N.), Schwartz, J.T.).- Linear Operators, Part I : General Theory, Interscience (New York), 1957. · Zbl 0084.10402
[4] Kelley, J.L.).- General topology, Van Nostrand Reinhold (New York), 1955 . · Zbl 0066.16604
[5] Moreau, J.J.).- Bounded variation in time, in : Topics in Nonsmooth Mechanics, (Ed. J.J. Moreau, P.D. Panagiotopoulos and G. Strang), Birkhäuser (Basel, Boston, Berlin), 1988, p. 1-74. · Zbl 0657.28008
[6] Moreau, J.J.).- Unilateral contact and dry friction in finite freedom dynamics, in : Non-smooth Mechanics and Applications, (Ed. J.J. Moreau and P.D. Panagiotopoulos), n°302, Springer-Verlag (Wien, New York), 1988, p. 1-82. · Zbl 0703.73070
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.