Abstract and Applied Analysis

Function Spaces with Bounded L p Means and Their Continuous Functionals

Massimo A. Picardello

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Abstract

This paper studies typical Banach and complete seminormed spaces of locally summable functions and their continuous functionals. Such spaces were introduced long ago as a natural environment to study almost periodic functions (Besicovitch, 1932; Bohr and Fölner, 1944) and are defined by boundedness of suitable L p means. The supremum of such means defines a norm (or a seminorm, in the case of the full Marcinkiewicz space) that makes the respective spaces complete. Part of this paper is a review of the topological vector space structure, inclusion relations, and convolution operators. Then we expand and improve the deep theory due to Lau of representation of continuous functional and extreme points of the unit balls, adapt these results to Stepanoff spaces, and present interesting examples of discontinuous functionals that depend only on asymptotic values.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2013), Article ID 609525, 26 pages.

Dates
First available in Project Euclid: 7 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412687022

Digital Object Identifier
doi:10.1155/2014/609525

Mathematical Reviews number (MathSciNet)
MR3173284

Citation

Picardello, Massimo A. Function Spaces with Bounded ${L}^{p}$ Means and Their Continuous Functionals. Abstr. Appl. Anal. 2014, Special Issue (2013), Article ID 609525, 26 pages. doi:10.1155/2014/609525. https://projecteuclid.org/euclid.aaa/1412687022


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References

  • W. Stepanoff, “Über einige Verallgemeinerungen der fast periodischen Funktionen,” Mathematische Annalen, vol. 95, no. 1, pp. 473–498, 1926.
  • H. Weyl, “Integralgleichungen und fastperiodische Funktionen,” Mathematische Annalen, vol. 97, no. 1, pp. 338–356, 1927.
  • A. S. Besicovitch, Almost Periodic Functions, Cambridge University Press, Cambridge, UK, 1932.
  • H. Bohr, “Zur Theorie der fastperiodischen čommentComment on ref. [7a?]: We split this reference to [7a,7b,7c?]. Please check.Funktionen, I,” Acta Mathematica, vol. 45, pp. 29–127, 1924.
  • H. Bohr, “Zur Theorie der fastperiodischen Funktionen, II,” Acta Mathematica, vol. 46, pp. 101–214, 1925.
  • H. Bohr, “Zur Theorie der fastperiodischen Funktionen, III,” Acta Mathematica, vol. 47, pp. 237–281, 1926.
  • K. S. Lau, “On the Banach spaces of functions with bounded upper means,” Pacific Journal of Mathematics, vol. 91, no. 1, pp. 153–172, 1980.
  • A. S. Besicovitch, “On generalized almost periodic functions,” Proceedings of the London Mathematical Society, vol. S2-25, no. 1, pp. 495–512.
  • J. Marcinkiewicz, “Un remarque sur les éspace de A.S. Besicovitch,” Comptes Rendus de l'Académie des Sciences, vol. 208, pp. 157–159, 1939.
  • H. Bohr and E. Fölner, “On some type of functional spaces,” Acta Mathematica, vol. 76, pp. 31–155, 1944.
  • J.-P. Bertrandias, “Espaces de fonctions bornées et continues en moyenne asymptotique d'ordre $p$,” Bulletin de la Société Mathématique de France, vol. 5, pp. 1–106, 1966.
  • K. S. Lau, “The class of convolution operators on the Marcinkiewicz spaces,” Annales de l'Institut Fourier, vol. 31, pp. 225–243, 1981.
  • F. Andreano and R. Grande, “Convolution on spaces of locally summable functions,” Journal of Function Spaces and Applications, vol. 6, no. 2, pp. 187–203, 2008.
  • F. Andreano and M. A. Picardello, “Approximate identities on some homogeneous Banach spaces,” Monatshefte für Mathematik, vol. 158, no. 3, pp. 235–246, 2009.
  • H. G. Feichtinger, “Banach convolution algebras of functions. II,” Monatshefte für Mathematik, vol. 87, no. 3, pp. 181–207, 1979.
  • H. G. Feichtinger, “An elementary approach to Wiener's third Tauberian theorem for the Euclidean $n$-space,” in Symposia Mathematica, vol. 29, pp. 267–301, Academic Press, New York, NY, USA, 1987.
  • K. S. Lau and J. K. Lee, “On generalized harmonic analysis,” Transactions of the American Mathematical Society, vol. 259, no. 1, pp. 75–97, 1980.
  • J. A. Clarkson, “Uniformly convex spaces,” Transactions of the American Mathematical Society, vol. 40, no. 3, pp. 396–414, 1936.
  • O. Hanner, “On the uniform convexity of \emphL $^{p}$ and l $^{p}$,” Arkiv för Matematik, vol. 3, pp. 239–244, 1956.
  • L. Gillman and M. Jerison, Rings of Continuous Functions, Graduate Texts in Mathematics, Springer, New York, NY, USA, 1976.
  • K. Yosida and E. Hewitt, “Finitely additive measures,” Transactions of the American Mathematical Society, vol. 72, pp. 46–66, 1952.
  • E. Alfsen and E. Effros, “Structure in real Banach spaces, part I,” Annals of Mathematics, vol. 96, pp. 98–128, 1972. \endinput