## Kyoto Journal of Mathematics

### On the coefficients of Vilenkin-Fourier series with small gaps

Bhikha Lila Ghodadra

#### Abstract

The Riemann-Lebesgue lemma shows that the Vilenkin-Fourier coefficient $\hat{f}(n)$ is of $o(1)$ as $n\rightarrow \infty$ 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

#### 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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