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Vector-valued integration in \(BK\)-spaces. (English) Zbl 0854.46018

Summary: Questions of convergence in \(BK\)-spaces, i.e. Banach spaces of complex-valued sequences \(x= (x_k )_{k\in \mathbb{Z}}\) with continuity of all functionals \(x\mapsto x_k\) \((k\in \mathbb{Z})\) will be studied by methods of Fourier analysis. An elegant treatment is possible if the Cesàro sections of a \(BK\)-space element \(x\) can be represented by vector-valued Riemann integrals. This was done by G. Goes [Stud. Math. 57, 241-249 (1976; Zbl 0346.42004)] following the example of Y. Katznelson [“An introduction to harmonic analysis” (1968; Zbl 0169.17902), pp. 10-12]. The purpose of this paper is to make precise the conditions in [Goes, loc. cit.] concerning Riemann integration and to demonstrate relations between \(BK\)-spaces which are generated by a given \(BK\)-space.

MSC:

46B45 Banach sequence spaces
46A45 Sequence spaces (including Köthe sequence spaces)
28B05 Vector-valued set functions, measures and integrals
40G05 Cesàro, Euler, Nörlund and Hausdorff methods
42A24 Summability and absolute summability of Fourier and trigonometric series
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References:

[1] Goes, G.: Generalizations of theorems of Fejér and Zygmund on convergence and bound - edness of conjugate series. Studia Math. 57 (1976), 241 - 249. · Zbl 0346.42004
[2] Gordon, R.: Riemann integration in Banach spaces. Rocky Mountain J. Math. 21(1991), 923 - 949. · Zbl 0764.28008 · doi:10.1216/rmjm/1181072923
[3] Hewitt, E. and K. A. Ross: Abstract Harmonic Analysis. Vol. 1: Structure of Topological Groups, Integration Theory, Group Representations. Berlin - Göttingen - Heidelberg: Springer-Verlag 1963. 15] Katznelson, Y.: An Introduction to Harmonic Analysis. New York: Wiley 1968. (6] Yosida, K.: Functional Analysis. Berlin - Gottingen - Heidelberg: Springer-Verlag 1965. 17] Zeller, K.: Abschnittskonvergenz in FK-Ràumen. Math. Z. 55 (1951), 55 - 70.
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