Special affine multiresolution analysis and the construction of orthonormal wavelets in L²(ℝ)

Abstract

The aim of this study is to construct both the continuous and discrete special affine wavelets in $L^2(\mathbb{R})$. Firstly, we introduce the continuous special affine wavelet transform via the convolution operations in the special affine Fourier domain. Subsequently, we investigate all the fundamental properties of the proposed transform using special affine Fourier transforms. Secondly, the discrete orthonormal special affine wavelets are constructed by introducing the notion of the special affine multiresolution analysis (MRA) in $L^2(\mathbb{R})$. Besides, a fast wavelet transform associated with special affine MRA is also presented. Towards the culmination, the formulation of special affine MRA starting from a given scaling function is briefly studied.

Keywords:

• Special affine MRA
• special affine wavelet
• filter
• scaling function
• special affine Fourier transform

2010 Mathematics Subject Classifications:

• 42C40
• 43A70
• 44A35
• 42C15

Acknowledgments

The authors would like to thank the anonymous referees for their careful reading and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author is supported by SERB (Science and Engineering Research Board) (DST), Government of India [Grant No. EMR/2016/007951].