# IJPAM: Volume 82, No. 3 (2013)

**ORDER OF MAGNITUDE OF MULTIPLE WALSH-FOURIER**

COEFFICIENTS OF FUNCTIONS OF BOUNDED -VARIATION

COEFFICIENTS OF FUNCTIONS OF BOUNDED -VARIATION

Department of Mathematics

Faculty of Science

The Maharaja Sayajirao University of Baroda

Vadodara, 390 002 (Gujarat), INDIA

**Abstract. **For a Lebesgue integrable complex-valued function defined over the -dimensional torus
, let
denote the multiple Walsh-Fourier coefficient of , where
,
. The Riemann-Lebesgue lemma shows that
as
for any
. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. When the definitive results are due to B. L. Ghodadra and J. R. Patadia [J. Inequal. Pure Appl. Math., 9 (2) (2008), Article 44] for functions of certain classes of functions of generalized bounded variation. Ghodadra [Acta Math. Hungar 128 (4), 2010, 328-343] defined the notion of bounded -variation () for a function from a rectangle
to and obtained definitive results for the order of magnitude of multiple trigonometric Fourier coefficients. In this paper, such definitive results for the order of magnitude of multiple Walsh-Fourier coefficients for a function of bounded -variation are obtained.

**Received: **September 4, 2012

**AMS Subject Classification: **42C10, 42B05, 26B30, 26D15

**Key Words and Phrases: **multiple Walsh-Fourier coefficient, function of bounded -variation in several variables, order of magnitude

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**Source:**International Journal of Pure and Applied Mathematics

**ISSN printed version:**1311-8080

**ISSN on-line version:**1314-3395

**Year:**2013

**Volume:**82

**Issue:**3