Biorthogonal wavelets on the spectrum. (20th November 2020)

• Record Type: Journal Article
• Authors: Ahmad, Owais; Sheikh, Neyaz A.; Nisar, Kottakkaran Sooppy; Shah, Firdous A.
• Is Part Of: Mathematical methods in the applied sciences. Volume 44:Number 6(2021)
• Page Start: 4479
• Page End: 4490
• Publication Date: 2020-11-20
• Subjects: Biorthogonal wavelet -- Fourier transform -- nonuniform multiresolution analysis -- spectrum -- wavelet; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.7046
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum Λ = 0, r / N + 2 ℤ, where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2 N − 1 such that r and N are relatively prime. We first establish the necessary and sufficient conditions for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families. Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.