Arkiv för Matematik

  • Ark. Mat.
  • Volume 20, Number 1-2 (1982), 101-109.

Absolute convergence of Fourier series on totally disconnected groups

Walter R. Bloom

Full-text: Open access

Abstract

LetG denote a totally disconnected locally compact metric abelian group with translation invariant metric d and character group ΓG. The Lipschitz spaces are defined by $Lip\left( {\alpha ;p} \right) = \left\{ {f \in L^p \left( G \right):\left\| {\tau _a f - f} \right\|_p = O\left( {d\left( {a,0} \right)^\alpha } \right),a \to 0} \right\},$ where τaf: xf(x-a) and α∈(0,1). For a suitable choice of metric it is shown that Lip (α; p)⊂Lr(ΓG), where α>1/p+1/r−1≧0 and 1≦p≦2. In the case G is compact the corresponding result holds for α>1/r−1/2 and p>2. In addition for G non-discrete the above result is shown to be sharp, in the sense that the range of values of α cannot be extended. The results include classical theorems of S. N. Bernstein, O. Szász and E. C. Titchmarsh.

Article information

Source
Ark. Mat., Volume 20, Number 1-2 (1982), 101-109.

Dates
Received: 1 January 1980
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485896972

Digital Object Identifier
doi:10.1007/BF02390501

Mathematical Reviews number (MathSciNet)
MR660128

Zentralblatt MATH identifier
0492.43004

Subjects
Primary: 43A25: Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
Secondary: 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 43A70: Analysis on specific locally compact and other abelian groups [See also 11R56, 22B05]

Rights
1982 © Institut Mittag Leffler

Citation

Bloom, Walter R. Absolute convergence of Fourier series on totally disconnected groups. Ark. Mat. 20 (1982), no. 1-2, 101--109. doi:10.1007/BF02390501. https://projecteuclid.org/euclid.afm/1485896972


Export citation

References

  • George Benke, Smoothness and absolute convergence of Fourier series in compact totally disconnected groups J. Funct. Anal. 29 (1978), 319–327.
  • George Benke, Trigonometric approximation theory in compact totally disconnected groups, Pacific J. Math. 77 (1978), 23–32.
  • Serge Bernstein, Sur la convergence absolue des séries trigonométriques, C.R. Acad. Sci. Paris 158 (1914), 1661–1663.
  • Walter R. Bloom, Bernstein's inequality for locally compact Abelian groups, J. Austral. Math. Soc. 17 (1974), 88–101.
  • Walter R. Bloom, Jackson's Theorem for locally compact Abelian groups, Bull. Austral. Math. Soc. 10 (1974), 59–66.
  • Walter R. Bloom, Jackson's Theorem for finite products and homomorphic images of locally compact Abelian groups, Bull. Austral. Math. Soc. 12 (1975), 301–309.
  • Walter R. Bloom, Absolute convergence of Fourier series on finite dimensional groups, Colloq. Math. (to appear).
  • John Scott Bradley, Interpolation theory and Lipschitz classes on totally disconnected groups, M. Sc. Thesis, The University of British Columbia, 1974.
  • N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414.
  • Colin C. Graham, The Sidon constant of a finite abelian group, Proc. Amer. Math. Soc. 68 (1978), 83–84.
  • Edwin Hewitt and Kenneth A. Ross, Abstract harmonic analysis, Vols. I, II, Die Grundlehren der mathematischen Wissenschaften, Bände 115, 152, Springer-Verlag, Berlin, Heidelberg, New York, 1963, 1970.
  • Jean-Pierre Kahane, Séries de Fourier absolument convergentes, Ergebnisse der Mathematik und ihrer Grenzgebiete, 50, Springer-Verlag, Berlin, Heidelberg, New York, 1970.
  • C. W. Onneweer, Absolute convergence of Fourier series on certain groups, Duke Math. J. 39 (1972), 599–609.
  • C. W. Onneweer, Absolute convergence of Fourier series on certain groups, II, Duke Math. J. 41 (1974), 679–688.
  • T. S. Quek and Leonard Y. H. Yap, Absolute convergence of Vilenkin-Fourier series, J. Math. Anal. Appl. 74 (1980), 1–14.
  • Otto Szász, Über den Konvergenzexponenten der Fourierschen Reihen gewisser Funktionenklassen, S.-B. Bayer. Akad. Wiss. Math.-Phys. Kl. 1922, 135–150.
  • Otto Szász, Über die Fourierschen Reihen gewisser Funktionenklassen, Math. Ann. 100 (1928), 530–536.
  • E. C. Titchmarsh, A note on Fourier transforms, J. London Math. Soc. 2 (1927), 148–150.
  • N. Ja. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 363–400; English transl., Amer. Math. Soc. Transl. (2)28 (1963), 1–35.
  • N. Ya. Vilenkin and A. I. Rubinshtein, A theorem of S. B. Stechkin on absolute convergence of a series with respect to systems of characters on zero-dimensional abelian groups, Izv. Vysš. Učebn. Zaved. Matematika 19 (1975), 3–9; English transl., Soviet Math. 19 (1976), 1–6.
  • P. L. Walker, Lipschitz classes on 0-dimensional groups, Proc. Cambridge Philos. Soc. 63 (1967), 923–928.
  • P. L. Walker, Lipschitz classes on finite dimensional groups, Proc. Cambridge Philos. Soc. 66 (1969), 31–38.