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Transactions of the American Mathematical Society

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On generalized harmonic analysis


Authors: Ka Sing Lau and Jonathan K. Lee
Journal: Trans. Amer. Math. Soc. 259 (1980), 75-97
MSC: Primary 42C99; Secondary 26A45, 43A32, 46E30
DOI: https://doi.org/10.1090/S0002-9947-1980-0561824-5
MathSciNet review: 561824
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Abstract: Motivated by Wiener's work on generalized harmonic analysis, we consider the Marcinkiewicz space $ {{\mathcal{M}}^p}({\textbf{R}})$ of functions of bounded upper average p power and the space $ {{\mathcal{V}}^p}({\textbf{R}})$ of functions of bounded upper p variation. By identifying functions whose difference has norm zero, we show that $ {{\mathcal{V}}^p}({\textbf{R}})$, $ 1\, < \,p\, < \infty $, is a Banach space. The proof depends on the result that each equivalence class in $ {{\mathcal{V}}^p}({\textbf{R}})$ contains a representative in $ {L^p}({\textbf{R}})$. This result, in turn, is based on Masani's work on helixes in Banach spaces. Wiener defined an integrated Fourier transformation and proved that this transformation is an isometry from the nonlinear subspace $ {{\mathcal{W}}^2}({\textbf{R}})$ of $ {{\mathcal{M}}^2}({\textbf{R}})$ consisting of functions of bounded average quadratic power, into the nonlinear subspace $ {{\mathcal{u}}^2}({\textbf{R}})$ of $ {{\mathcal{V}}^2}({\textbf{R}})$ consisting of functions of bounded quadratic variation. By using two generalized Tauberian theorems, we prove that Wiener's transformation W is actually an isomorphism from $ {{\mathcal{M}}^2}({\textbf{R}})$ onto $ {{\mathcal{V}}^2}({\textbf{R}})$. We also show by counterexamples that W is not an isometry on the closed subspace generated by $ {{\mathcal{W}}^2}({\textbf{R}})$.


References [Enhancements On Off] (What's this?)

  • [1] J. Bertrandias, Espaces de fonctions bornés et continues en moyenne asymptotique d'ordre p, Bull. Soc. Math. France 5 (1966).
  • [2] R. Boas, Functions which are odd about several points, Nieuw Arch. Wisk. 1 (1953), 27-32. MR 0054836 (14:987e)
  • [3] H. Bohr and E. Følner, On some types of functional spaces, Acta Math. 76 (1945), 31-155. MR 0013443 (7:154f)
  • [4] F. Carroll, Functions whose differences belong to $ {L^p}[0,\,1]$, Indag. Math. 26 (1964), 250-255. MR 0170993 (30:1226)
  • [5] H. Ellis and J. Halperin, Function spaces determined by a levelling length function, Canad. J. Math. 5 (1953), 576-592. MR 0058869 (15:439c)
  • [6] G. Hardy and J. Littlewood, Some properties of fractional integrals, Math. Z. 27 (1928), 565-606. MR 1544927
  • [7] E. Hewitt and K. Ross, Abstract harmonic analysis. I, II, Springer-Verlag, Berlin, 1963, 1970. MR 0262773 (41:7378)
  • [8] E. Hille and R. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
  • [9] K. Lau, On the Banach spaces of functions with bounded upper means, Pacific J. Math. (to appear). MR 612895 (83b:46037)
  • [10] J. Lee, On a class of functions in generalized harmonic analysis, Notices Amer. Math. Soc. 170 (1970), 634. Abstract 674-106.
  • [11] -, The completeness of the class of functions of bounded upper p-variation, $ 1\, < \,p\, < \,\infty $, Notices Amer. Math. Soc. 17 (1970), 1057. Abstract 681-B5.
  • [12] W. Luxemburg and A. Zaanen, Notes on Banach function spaces. I, Indag. Math. 25 (1963), 135-147. MR 0149231 (26:6723a)
  • [13] J. Marcinkiewicz, Une remarque sur les espaces de M. Besicovitch, C. R. Acad. Sci. Paris 208 (1939), 157-159.
  • [14] P. Masani, On helixes in Banach spaces, Sānkhya 38 (1976), 1-27. MR 0471051 (57:10792)
  • [15] -, An outline of vector graph and conditional Banach spaces, Linear Space and Approximation (P. Butzer and B. Sz.-Nagy, eds.) Birkhäuser-Verlag, Basel, 1978, pp. 72-89.
  • [16] -, Commentary on the memoire on generalized harmonic analysis [30a], Norbert Wiener: Collected Work, Vol. II, P. Masani, ed. (to appear).
  • [17] R. Nelson, The spaces of functions of finite upper p-variation, Trans. Amer. Math. Soc. 253 (1979), 171-190. MR 536941 (80i:46027)
  • [18] N. Wiener, Generalized harmonic analysis, Acta Math. 55 (1930), 117-258. MR 1555316
  • [19] -, The Fourier integral and certain of its application, Dover, New York, 1959. MR 0100201 (20:6634)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0561824-5
Keywords: Banach spaces, generalized harmonic analysis, helixes, Marcinkiewicz spaces, Tauberian theorem, upper p-variation, Wiener transformation
Article copyright: © Copyright 1980 American Mathematical Society

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