Wavelets on the Spectrum

Shukla, Niraj K.; Mittal, Shiva

doi:10.1080/01630563.2013.848366

Cited by 8 publications

(3 citation statements)

References 8 publications

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“…Shah and Abdullah 11 established NUMRA on non‐Archimedean local fields. The development of biorthogonal wavelets associated with MRA and NUMRA were carried by various researchers 1,12‐14 …”

Section: Introductionmentioning

confidence: 99%

Biorthogonal wavelets on the spectrum

Ahmad

Sheikh

Nisar

et al. 2020

Math Methods in App Sciences

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In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum normalΛ={}0,rfalse/N+2ℤ, where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2N − 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.

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Section: Introductionmentioning

confidence: 99%

Biorthogonal wavelets on the spectrum

Ahmad

Sheikh

Nisar

et al. 2020

Math Methods in App Sciences

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“…A parallel development on local fields of positive characteristic was carried by Shah and his colleagues [7,8]. The concept of biorthogonal wavelets associated with MRA and NUMRA were studied by various researchers [1,6,9].…”

Section: Introductionmentioning

confidence: 99%

Biorthogonal Wavelets on the Spectrum

Ahmad¹,

Shah²

2017

Preprint

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A generalization of Mallat's classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed (J. Funct. Anal. 158, 209-241, 1998). In this article, we introduce the notion of biorthgonoal nonuniform multiresolution analysis on the spectrum Λ = {0, r/N } + 2Z, where N ≥ 1 is an integer and r is an odd integer with 1 ≤ r ≤ 2N −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.

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“…The main contribution of this paper is that we extend our previous work [7] and construct a class of J th -stage nonuniform discrete wavelet systems on local fields of positive characteristic. Different from our previous approach in the orthonormal case, our analysis of nonuniform discrete scheme is inspired by Shukla and Mittal's approach in [12] for construction of wavelets on the spectrum. We provide some characterizations of the J th -stage discrete wavelet systems to be orthonormal bases for the Hilbert space l 2 (Λ).…”

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confidence: 99%

Construction of $J^{\text{th}}$-stage Nonuniform Wavelets on Local Fields

Ahmad¹,

Shah²

2018

Preprint

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Shah and Abdullah [Complex Analysis Operator Theory, 9 (2015), [1589][1590][1591][1592][1593][1594][1595][1596][1597][1598][1599][1600][1601][1602][1603][1604][1605][1606][1607][1608] have introduced a generalized notion of nonuniform multiresolution analysis (NUMRA) on local field K of positive characteristic in which the translation set Λ acting on the scaling function to generate the core space V 0 is no longer a group, but is the union of Z and a translate of Z, given by Λ = {0, u(r)/N } + Z, where N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime, andIn this paper, we focus on the extension of nonuniform continuous wavelets to the construction of J th -stage nonuniform discrete wavelets on local fields. We establish some general characterizations for the J th -stage nonuniform discrete wavelet systems to be orthornormal bases in L 2 (Λ). Moreover, we establish a relation between the continuous wavelets of L 2 (K) and their discrete counterparts of l 2 (Λ).

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Wavelets on the Spectrum

Cited by 8 publications

References 8 publications

Biorthogonal wavelets on the spectrum

Biorthogonal wavelets on the spectrum

Biorthogonal Wavelets on the Spectrum

Construction of $J^{\text{th}}$-stage Nonuniform Wavelets on Local Fields

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