Kyoto Journal of Mathematics

On the coefficients of Vilenkin-Fourier series with small gaps

Bhikha Lila Ghodadra

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 10 November 2011

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1320936737

Digital Object Identifier
doi:10.1215/21562261-1424902

Mathematical Reviews number (MathSciNet)
MR2854157

Zentralblatt MATH identifier
1229.42031

Subjects
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

Citation

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. https://projecteuclid.org/euclid.kjm/1320936737


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