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The continuous wavelet transform and window functions

Authors: J. N. Pandey and S. K. Upadhyay
Journal: Proc. Amer. Math. Soc. 143 (2015), 4759-4773
MSC (2010): Primary 46F12; Secondary 46F05, 46F10
Published electronically: July 24, 2015
MathSciNet review: 3391034
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Abstract: We define a window function $ \psi $ as an element of $ L^2(\mathbb{R}^n)$ satisfying certain boundedness properties with respect to the $ L^2(\mathbb{R}^n)$ norm and prove that it satisfies the admissibility condition if and only if the integral of $ \psi (x_1,x_2,\cdots ,x_n)$ with respect to each of the variables $ x_1,x_2,\cdots ,x_n$ along the real line is zero. We also prove that each of the window functions is an element of $ L^1(\mathbb{R}^n)$. A function $ \psi \in L^2(\mathbb{R}^n)$ satisfying the admissibility condition is a wavelet. We define the wavelet transform of $ f\in L^2(\mathbb{R}^n)$ (which is a window function) with respect to the wavelet $ \psi \in L^2(\mathbb{R}^n)$ and prove an inversion formula interpreting convergence in $ L^2(\mathbb{R}^n)$. It is also proved that at a point of continuity of $ f$ the convergence of our wavelet inversion formula is in a pointwise sense.

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Additional Information

J. N. Pandey
Affiliation: School of Mathematics and Statistics, Carleton University, Ottawa, Canada

S. K. Upadhyay
Affiliation: Department of Mathematical Sciences, Indian Institute of Technology, DST-CIMS Banaras Hindu University, India

Keywords: Continuous wavelet transform, Fourier inversion theory, inverse wavelet transform
Received by editor(s): April 17, 2014
Received by editor(s) in revised form: July 2, 2014
Published electronically: July 24, 2015
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society