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Biorthogonal wavelets on the spectrum
Author(s) -
Ahmad Owais,
Sheikh Neyaz A.,
Nisar Kottakkaran Sooppy,
Shah Firdous A.
Publication year - 2021
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.7046
Subject(s) - biorthogonal system , wavelet , mathematics , multiresolution analysis , scaling , integer (computer science) , prime (order theory) , spectrum (functional analysis) , function (biology) , biorthogonal wavelet , characterization (materials science) , pure mathematics , mathematical analysis , wavelet transform , combinatorics , geometry , discrete wavelet transform , computer science , quantum mechanics , artificial intelligence , evolutionary biology , materials science , nanotechnology , physics , biology , programming language
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.