Construction of periodic wavelet frames generated by the Walsh polynomials (Q901469)

Recently, \textit{F. A. Shah} [Int. J. Wavelets Multiresolut. Inf. Process. 11, No. 6, Article ID 1350042, 15 p. (2013; Zbl 1288.42013)] established a unitary extension principle for constructing normalized tight wavelet frames generated by the Walsh polynomials on \(\mathbb{R}^+\). Using this result, the authors 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 \(\mathbb{R}^+\). Furthermore, based on the mixed extension principle and Walsh-Fourier transforms of the wavelet frames, an explicitly constructed method for a pair of dual periodic wavelet frames generated by the Walsh polynomials is also given.