Kyoto Journal of Mathematics

On the coefficients of Vilenkin-Fourier series with small gaps

Bhikha Lila Ghodadra

Full-text: Open access


The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient (n) is of o(1) as n for any integrable function f on Vilenkin groups. However, it is known that the Vilenkin-Fourier coefficients of integrable functions can tend to zero as slowly as we wish. The definitive result is due to B. L. Ghodadra for functions of certain classes of generalized bounded fluctuations. We prove that this is a matter only of local fluctuation for functions with the Vilenkin-Fourier series lacunary with small gaps. Our results, as in the case of trigonometric Fourier series, illustrate the interconnection between ‘localness’ of the hypothesis and type of lacunarity and allow us to interpolate the results.

Article information

Kyoto J. Math., Volume 51, Number 4 (2011), 891-900.

First available in Project Euclid: 10 November 2011

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
Secondary: 26D15: Inequalities for sums, series and integrals 43A40: Character groups and dual objects 43A75: Analysis on specific compact groups


Ghodadra, Bhikha Lila. On the coefficients of Vilenkin-Fourier series with small gaps. Kyoto J. Math. 51 (2011), no. 4, 891--900. doi:10.1215/21562261-1424902.

Export citation


  • [1] H. E. Chrestenson, A class of generalized Walsh functions, Pacific J. Math. 5 (1955), 17–31.
  • [2] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372–414.
  • [3] B. L. Ghodadra, On the magnitude of Vilenkin-Fourier coefficients, J. Indian Math. Soc. (N.S.) 75 (2008), 93–104.
  • [4] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, Berlin, 1979.
  • [5] C. W. Onneweer, Absolute convergence of Fourier series on certain groups, Duke Math. J. 39 (1972), 599–609.
  • [6] C. W. Onneweer and D. Waterman, Uniform convergence of Fourier series on groups, I, Michigan Math. J. 18 (1971), 265–273.
  • [7] C. W. Onneweer and D. Waterman, Fourier series of functions of harmonic bounded fluctuation on groups, J. Analyse Math. 27 (1974), 79–83.
  • [8] J. R. Patadia and R. G. Vyas, Fourier series with small gaps and functions of generalised variations, J. Math. Anal. Appl. 182 (1994), 113–126.
  • [9] J. R. Patadia and J. H. Williamson, Local theorems for the absolute convergence of lacunary Fourier series on certain groups, Quart. J. Math. Oxford Ser. (2) 37 (1986), 81–93.
  • [10] R. E. A. C. Paley, A remarkable series of orthogonal functions, Proc. London Math. Soc. 34 (1932), 241–279.
  • [11] M. Pepić, About characters and the Dirichlet kernel on Vilenkin groups, Sarajevo J. Math. 4 (16) (2008), 109–123.
  • [12] M. Schramm and D. Waterman, Absolute convergence of Fourier series of functions of ΛBV(p) and ϕΛBV, Acta. Math. Acad. Sci. Hungar. 40 (1982), 273–276.
  • [13] M. Schramm and D. Waterman, On the magnitude of Fourier coefficients, Proc. Amer. Math. Soc. 85 (1982), 407–410.
  • [14] N. J. Vilenkin, On a class of complete orthonormal systems, Bull. Acad. Sci. URSS Ser. Math. [Izvestia Akad. Nauk SSSR] 11 (1947), 363–400; Amer. Math. Soc. Transl. (2) 28 (1963), 1–35.