IJPAM: Volume 82, No. 3 (2013)

ORDER OF MAGNITUDE OF MULTIPLE WALSH-FOURIER
COEFFICIENTS OF FUNCTIONS OF BOUNDED $p$-VARIATION

Bhikha Lila Ghodadra
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 $f$ defined over the $n$-dimensional torus $\mathbb {I}^n:=[0,1)^n$ $(n\in \mathbb N)$, let $\hat f({\bf k})$ denote the multiple Walsh-Fourier coefficient of $f$, where ${\bf k}=(k_1,\dots,k_n)\in (\mathbb {Z}^+)^n$, $\mathbb {Z^+}=\mathbb {N}\cup \{0\}$. The Riemann-Lebesgue lemma shows that $\hat f({\bf k})=o(1)$ as $\vert{\bf k}\vert\to 0$ for any $f\in {\rm L}^1(\mathbb I^n)$. However, it is known that, these Fourier coefficients can tend to zero as slowly as we wish. When $n=1$ 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 $p$-variation ($p\ge 1$) for a function from a rectangle $[a_1,b_1]\times\cdots \times [a_n,b_n]$ to $\mathbb {C}$ 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 $p$-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 $p$-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