Construction of Periodic Wavelet Frames Generated by the Walsh Polynomials

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.