IJPAM: Volume 114, No. 3 (2017)

Title

THE POINT WISE BEHAVIOR OF 2-DIMENSIONAL
WAVELET EXPANSIONS IN $L^p(R^2)$

Authors

Raghad Sahib Shamsah$^{1,2}$, Anvarjon A. Ahmedov$^3$, Hishamuddin Zainuddin$^1$, Adem Kilicman$^4$, Fudziah Ismail$^4$
$^1$Institute for Mathematical Research
Universiti Putra Malaysia
43400 UPM, Serdang, Selangor, MALAYSIA
$^2$Department of Mathematics
Faculty of Education for Pure Sciences
Universiti of Karbala, IRAQ
$^3$Faculty of Industrial Science and Technology
Universiti Pahang Malaysia
26300 UMP, Lebuhraya Tun razak, Pahang, MALAYSIA
$^4$Department of Mathematics
Faculty of Science
Universiti Putra Malaysia
43400 UPM, Serdang, Selangor, MALAYSIA

Abstract

We show that the two dimensional wavelet expansion of $L^p(R^2)$ function for $1 <p< \infty$ converges pointwise almost everywhere under wavelet projection operator. This convergence can be established by assuming some minimal regularity to get the rapidly decreasing for two dimensional wavelet $ \psi_{j_{1},j_{2},k_{1},k_{2}}$. The Kernel function of the wavelet projection operator in two dimension converges absolutely, distributionally and is bounded. Also the wavelet expansions in two dimension are controlled in a magnitude by the maximal function operator. All these conditions can be utilized to achieve the convergence almost everywhere.

History

Received: December 14, 2016
Revised: March 15, 2017
Published: May 23, 2017

AMS Classification, Key Words

AMS Subject Classification: 40A05, 42C40, 45P05, 47B38, 46E27, 28A33
Key Words and Phrases: two dimensional wavelet expansion, kernel function, almost everywhere convergence, maximal function, bounded

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How to Cite?

DOI: 10.12732/ijpam.v114i3.9 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2017
Volume: 114
Issue: 3
Pages: 523 - 536


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