Abstract
An explicit method for the construction of a tight wavelet frame generated by the Walsh polynomials with the help of extension principles was presented by Shah (Shah, 2013). In this article, we extend the notion of wavelet frames to periodic wavelet frames generated by the Walsh polynomials on R+ by using extension principles. We first show that under some mild conditions, the periodization of any wavelet frame constructed by the unitary extension principle is still a periodic wavelet frame on R + . Then, we construct a pair of dual periodic wavelet frames generated by the Walsh polynomials on R + using the machinery of the mixed extension principle and Walsh–Fourier transforms.
Highlights
Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science
The concept of multiresolution analysis on a positive half-line R+ was recently introduced by Farkov [13]
We prove that under some mild conditions, the periodization of any wavelet frame constructed by the unitary extension principle is a periodic wavelet frame on a positive half-line
Summary
Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. One of the Mathematics 2015, 3 most useful methods to construct wavelet frames is through the concept of the unitary extension principle (UEP) introduced by Ron and Shen [1] and was subsequently extended by Daubechies et al [2] in the form of the oblique extension principle (OEP) They give sufficient conditions for constructing tight wavelet frames for any refinable function φ(x) that generates a multiresolution analysis. The concept of multiresolution analysis on a positive half-line R+ was recently introduced by Farkov [13] He pointed out a method for constructing compactly-supported orthogonal p-wavelets related to the Walsh functions and proved necessary and sufficient conditions for scaling filters with pn many terms (p, n ≥ 2) to generate a p-MRAin L2 (R+ ).
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