On some class of nonseparable continuous wavelet transforms

Abstract

It is well-known that only a single condition (called the admissibility condition) is sufficient for L ₂-convergence of multiple continuous wavelet transforms (MCWT). However known results suggest that to guarantee the pointwise convergence of MCWT for L _p -functions wavelets should vanish quite rapidly at infinity. In this article, we consider the class of nonseparable multiple wavelets with square-symmetric Fourier transforms. For these wavelets ψ we prove that the admissibility condition and the condition ψ∈L ₁(R ⁿ ) are sufficient for the convergence of corresponding MCWT almost everywhere.

Keywords:

• multidimensional continuous wavelet transforms
• convergence in norm and pointwise

AMS Subject Classifications::

• Primary 42C15
• Secondary 40A30

Acknowledgements

This article has been supported by the Ministry of Higher Education of Malaysia under Research Grant (FRGS, Fasa 2/2009, Grant Number 01-11-09-713FR/F1). The first author expresses his gratitude to University Putra Malaysia for support under IGRF scheme. The authors are grateful to Applicable Analysis referees for their comments which helped to improve the quality of this article.