Nonuniform wavelet packets on local fields of positive characteristic

Shah, Firdous A.; Bhat, M. Younus

doi:10.2298/fil1706491s

Cited by 13 publications

(5 citation statements)

References 20 publications

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“…These studies were continued by Gabardo and his colleagues in previous works, 3–5 wherein they derived an extension of Cohen's theorem which gives the necessary and sufficient condition for the orthonormality of the collection

({}, ϕ false(\cdot - λ false) : λ \in normalΛ)

and provided a complete characterization of associated wavelets by means of its dimension function. The theory of nonuniform wavelets were further studied and investigated by several researchers in different directions, for instance, biorthogonal nonuniform wavelets, 6 characterization of scaling functions associated with an NUMRA, 7 nonuniform wavelet packets, 8 nonuniform wavelet frames (see Dai et al 9 and Debnath 10 ), nonuniform wavelets and wavelet packets on local fields of positive characteristic (see previous works 11–13 ) and vector‐valued nonuniform wavelets and wavelet packets (see previous works 14,15 ).…”

Section: Introductionmentioning

confidence: 99%

Fractional nonuniform multiresolution analysis in L2(ℝ)

Srivástava

Shah

Lone

2021

Math Methods in App Sciences

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To provide a significantly richer representation of non‐stationary signals appearing in various disciplines of science and engineering, we introduce a novel fractional nonuniform multiresolution analysis (FrNUMRA) on the spectrum Λ given by normalΛ={}0,rN+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. The necessary and sufficient condition for the existence of nonuniform wavelets of fractional order is derived and an algorithm is also presented for the construction of fractional NUMRA starting from a fractional low‐pass filter h0α with appropriate conditions. Besides, we provide a complete characterization for the biorthogonality of the translates of the scaling functions of two fractional nonuniform multiresolution analyses and the associated fractional biorthogonal wavelet families.

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({}, ϕ false(\cdot - λ false) : λ \in normalΛ)

Section: Introductionmentioning

confidence: 99%

Fractional nonuniform multiresolution analysis in L2(ℝ)

Srivástava

Shah

Lone

2021

Math Methods in App Sciences

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“…These studies were proceeded by Gabardo and his colleagues in [10,11,28], wherein they establish an extension of Cohen's theorem which provides the necessary and sufficient condition for the orthonormality of the system {φ(• − λ) : λ ∈ Ω} and presented some equivalent conditions of the associated wavelets via dimension functions. The theory of nonuniform wavelets was further studied and extensively investigated by many researchers in different directions, for example, nonuniform wavelet packets [2], vector-valued nonuniform wavelet packets [15], generalized nonuniform MRA [16], nonuniform wavelet frames [18,23], nonuniform wavelets and wavelet packets on local fields of positive characteristic [19,20,21].…”

Section: Introductionmentioning

confidence: 99%

Nonuniform Multiresolution Analysis Associated with Linear Canonical Transform

Shah¹,

Lone²

2020

Preprint

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The linear canonical transform (LCT) has attained respectable status within a short span and is being broadly employed across several disciplines of science and engineering including signal processing, optical and radar systems, electrical and communication systems, quantum physics etc, mainly due to the extra degrees of freedom and simple geometrical manifestation. In this article, we introduce a novel multiresolution analysis (LCT-NUMRA) on the spectrum Ω = 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, by intertwining the ideas of nonuniform MRA and linear canonical transforms. We first develop nonuniform multiresolution analysis associated with the LCT and then drive an algorithm to construct an LCT-NUMRA starting from a linear canonical low-pass filter Λ M 0 (ω) with suitable conditions. Nevertheless, to extend the scope of the present study, we construct the associated wavelet packets for such an MRA and investigate their properties by means of the linear canonical transforms.

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“…Sharma and Manchanda [15] presented a necessary and sufficient conditions for nonuniform wavelet frames in the frequency domain. The theory of nonuniform wavelets was further studied and investigated by several authors in different directions including wavelets, vector-valued wavelets and wavelet packets on local fields of positive characteristic [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Inequalities for nonuniform wavelet frames

Shah

2019

Georgian Mathematical Journal

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Gabardo and Nashed have studied nonuniform wavelets based on the theory of spectral pairs for which the associated translation set Λ = {0, r/N } + 2 Z is no longer a discrete subgroup of R but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we construct the associated wavelet frames and establish some sufficient conditions that ensure the nonuniform wavelet system ψ j,λ (x) = (2N ) j/2 ψ (2N ) j x − λ , j ∈ Z, λ ∈ Λ to be a frame for L 2 (R). The conditions proposed are stated in terms of the Fourier transforms of the wavelet system's generating functions.

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Nonuniform wavelet packets on local fields of positive characteristic

Cited by 13 publications

References 20 publications

Fractional nonuniform multiresolution analysis in L2(ℝ)

Fractional nonuniform multiresolution analysis in L2(ℝ)

Nonuniform Multiresolution Analysis Associated with Linear Canonical Transform

Inequalities for nonuniform wavelet frames

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