Nonuniform nonstationary wavelets and associated multiresolution analysis on local fields

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) 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 characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

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A Short Note on Wavelet Frames Based on FMRA on Local Fields

Journal of Mathematics ◽

10.1155/2020/3957064 ◽

2020 ◽

Vol 2020 ◽

pp. 1-5

Author(s):

M. Younus Bhat

Keyword(s):

Multiresolution Analysis ◽

Positive Characteristic ◽

Minimum Energy ◽

Short Note ◽

Explicit Construction ◽

Local Fields ◽

Wavelet Frame ◽

Wavelet Frames ◽

Frame Multiresolution Analysis ◽

Field Of Positive Characteristic

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

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Vector-valued nonuniform multiresolution analysis on local fields

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691315500290 ◽

2015 ◽

Vol 13 (04) ◽

pp. 1550029 ◽

Cited By ~ 3

Author(s):

Firdous Ahmad Shah ◽

M. Younus Bhat

Keyword(s):

Multiresolution Analysis ◽

Orthonormal Basis ◽

Positive Characteristic ◽

Local Fields ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Vilenkin Groups ◽

Refinement Mask ◽

Necessary And Sufficient ◽

Vector Valued

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

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