scholarly journals Construction of nonuniform periodic wavelet frames on non-Archimedean fields

2020 ◽  
Vol 74 (2) ◽  
pp. 1
Author(s):  
Owais Ahmad
Keyword(s):  
Real Life ◽  
Discrete Subgroup ◽  
Extension Principle ◽  
Mathematical Tool ◽  
Wavelet Frames ◽  
Spectral Pair ◽  
Periodic Wavelet ◽  
Spectral Pairs ◽  
Unitary Extension ◽  
Periodic Wavelet Frames

In real life applications not all signals are obtained by uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. Gabardo and Nashed, and Gabardo and Yu filled this gap by the concept of nonuniform multiresolution analysis and nonuniform wavelets based on the theory of spectral pairs for which the associated translation set \(\Lambda= \{0,r/N\}+2\mathbb{Z}\) 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 introduce a notion of nonuniform periodic wavelet frame on non-Archimedean field. Using the Fourier transform technique and the unitary extension principle, we propose an approach for the construction of nonuniform periodic wavelet frames on non-Archimedean fields.

The Study of Tight Periodic Wavelet Frames and Wavelet Frame Packets and Applications

2014 ◽  
Vol 977 ◽  
pp. 532-535
Author(s):  
Qing Jiang Chen ◽  
Yu Zhou Chai ◽  
Chuan Li Cai
Keyword(s):  
Information Science ◽  
Approximation Order ◽  
Tight Frame ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Decomposition Scheme ◽  
Periodic Wavelet ◽  
Unitary Extension ◽  
Periodic Wavelet Frames ◽  
Necessary And Sufficient

Information science focuses on understanding problems from the perspective of the stake holders involved and then applying information and other technologies as needed. A necessary and sufficient condition is identified in term of refinement masks for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approxi-mation order of truncated tight frame series is established, which facilitates construction of tight periodic wavelet frames with desirable approximation order. The pyramid decomposition scheme is derived based on the generalized multiresolution structure.

scholarly journals ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

2021 ◽  
Vol 7 (1) ◽  
pp. 3
Author(s):  
Ishtaq Ahmed ◽  
Owias Ahmad ◽  
Neyaz Ahmad Sheikh
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Function ◽  
Real Life ◽  
Discrete Subgroup ◽  
Local Fields ◽  
Mathematical Tool ◽  
Spectral Pair ◽  
Spectral Pairs ◽  
The Given

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.

scholarly journals Inequalities for nonuniform wavelet frames

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Firdous A. Shah
Keyword(s):  
Fourier Transforms ◽  
Sufficient Conditions ◽  
Discrete Subgroup ◽  
Wavelet Frames ◽  
Wavelet System ◽  
One Dimensional ◽  
Spectral Pair ◽  
Spectral Pairs

Abstract Gabardo and Nashed studied nonuniform wavelets by using the theory of spectral pairs for which the translation set {\Lambda=\{0,r/N\}+2\mathbb{Z}} is no longer a discrete subgroup of {\mathbb{R}} but a spectrum associated with a certain one-dimensional spectral pair. In this paper, we establish three sufficient conditions for the nonuniform wavelet system {\{\psi_{j,\lambda}(x)=(2N)^{j/2}\psi((2N)^{j}x-\lambda),\,j\in\mathbb{Z},\,% \lambda\in\Lambda\}} to be a frame for {L^{2}(\mathbb{R})} . The proposed inequalities are stated in terms of Fourier transforms and hold without any decay assumptions on the generator of such a system.

On Parseval Wavelet Frames with Two or Three Generators via the Unitary Extension Principle

2014 ◽  
Vol 57 (2) ◽  
pp. 254-263 ◽  
Author(s):  
Ole Christensen ◽  
Hong Oh Kim ◽  
Rae Young Kim
Keyword(s):  
Extension Principle ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Wavelet System ◽  
Number Of Generators ◽  
Unitary Extension

AbstractThe unitary extension principle (UEP) by A. Ron and Z. Shen yields a sufficient condition for the construction of Parseval wavelet frames with multiple generators. In this paper we characterize the UEP-type wavelet systems that can be extended to a Parseval wavelet frame by adding just one UEP-type wavelet system. We derive a condition that is necessary for the extension of a UEP-type wavelet system to any Parseval wavelet frame with any number of generators and prove that this condition is also sufficient to ensure that an extension with just two generators is possible.

scholarly journals Unitary Extension Principle for Nonuniform Wavelet Frames in L2(ℝ)

2021 ◽  
Vol 17 (1) ◽  
pp. 79-94
Author(s):  
Hari Krishan Malhotra ◽  
◽  
Lalit Kumar Vashisht ◽  
Keyword(s):  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension

scholarly journals Polyphase matrix characterization of framelets on local fields of positive characteristic

2017 ◽  
Vol 9 (1) ◽  
pp. 248-259
Author(s):  
F. A. Shah ◽  
M. Y. Bhat
Keyword(s):  
Positive Characteristic ◽  
Local Fields ◽  
Extension Principle ◽  
Wavelet Frames ◽  
Unitary Extension ◽  
Polyphase Matrix

AbstractAn important tool for the construction of framelets on local fields of positive characteristic using unitary extension principle was presented by Shah and Debnath [Tight wavelet frames on local fields, Analysis, 33 (2013), 293-307]. In this article, we continue the study of framelets on local fields and present a polyphase matrix characterization of framelets generated by the extension principle.

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