Nonuniform wavelet frames in L2(ℝ)

2015 ◽  
Vol 08 (02) ◽  
pp. 1550034 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda
Keyword(s):  
Multiresolution Analysis ◽  
Sufficient Condition ◽  
Wavelet Frame ◽  
Necessary And Sufficient Condition ◽  
Wavelet Frames ◽  
Spectral Pairs ◽  
Necessary And Sufficient ◽  
Funct Anal

Gabardo and Nashed, [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241] introduced the nonuniform multiresolution analysis (NUMRA) whose translation set is not necessarily a group. The translation set is taken for elements in [Formula: see text], N ≥ 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ is the set of all integers. In this paper, we construct wavelet frame over the translation set Λ on L2(ℝ). We call it nonuniform wavelet frame. We establish necessary and sufficient condition for such wavelet frame. An example is presented at the end.

Necessary condition and sufficient conditions for nonuniform wavelet frames in L2(K)

2018 ◽  
Vol 16 (01) ◽  
pp. 1850005 ◽  
Author(s):  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Sufficient Conditions ◽  
Local Fields ◽  
Necessary Condition ◽  
Wavelet Basis ◽  
Wavelet Frame ◽  
Wavelet Frames ◽  
Constructive Algorithm ◽  
Spectral Pairs ◽  
Funct Anal

A constructive algorithm based on the theory of spectral pairs for constructing nonuniform wavelet basis in [Formula: see text] was considered by Gabardo and Nashed [Nonuniform multiresolution analysis and spectral pairs, J. Funct. Anal. 158 (1998) 209–241]. In this setting, the associated translation set is a spectrum [Formula: see text] which is not necessarily a group nor a uniform discrete set, given [Formula: see text] where [Formula: see text] (an integer) and [Formula: see text] is an odd integer with [Formula: see text] such that [Formula: see text] and [Formula: see text] are relatively prime and [Formula: see text] is the set of all integers. The objective of this paper is to construct nonuniform wavelet frame on local fields. A necessary condition and four sufficient conditions for nonuniform wavelet frame on local fields are given.

scholarly journals THE MODULUS OF WEAKLY COMPACT MULTIPLIERS ON THE BANACH ALGEBRA L1(𝒢)** OF A LOCALLY COMPACT GROUP

2012 ◽  
Vol 86 (2) ◽  
pp. 315-321
Author(s):  
MOHAMMAD JAVAD MEHDIPOUR
Keyword(s):  
Banach Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Locally Compact ◽  
Weakly Compact ◽  
Necessary And Sufficient ◽  
Funct Anal

AbstractIn this paper we give a necessary and sufficient condition under which the answer to the open problem raised by Ghahramani and Lau (‘Multipliers and modulus on Banach algebras related to locally compact groups’, J. Funct. Anal. 150 (1997), 478–497) is positive.

Vector-valued nonuniform multiresolution analysis on local fields

2015 ◽  
Vol 13 (04) ◽  
pp. 1550029 ◽  
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.

The Fine Characters of a Sort of Biorthogonal Four-Dimensional Nonseparable Wavelet Packs

2010 ◽  
Vol 159 ◽  
pp. 7-12
Author(s):  
Hong Lin Guo ◽  
Yu Min Yu
Keyword(s):  
Iteration Method ◽  
Wavelet Packets ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Wavelet Frames ◽  
New Approach ◽  
Decomposition Scheme ◽  
Necessary And Sufficient

In this article, the notion of orthogonal nonseparable four-dimensional wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is introduced. A new approach for constructing them is presented by iteration method and wavelets as well wavelet frames. The biorthogonality properties of four-dimensi- -onal wavelet packets are discussed. Three biorthogonality formulas concerning these wavelet packs are estabished. A necessary and sufficient condition for the existence of the pyramid decomposition scheme of space is presented.

A New Procedure for Designing a Kind of Orthogonal Vector-Valued Multiscaled Wavelets and Wavelet Wraps

2011 ◽  
Vol 204-210 ◽  
pp. 1733-1736
Author(s):  
Hong Wei Gao
Keyword(s):  
Multiresolution Analysis ◽  
Filter Bank ◽  
Matrix Theory ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Orthogonal Vector ◽  
Novel Method ◽  
Necessary And Sufficient ◽  
Vector Valued

In this paper, notion of vector-valued multiresolution analysis is introduced. So does the notion of orthogonal vector-valued wavelets A necessary and sufficient condition on the existence of orthogonal vector-valued wavelets is presented by using paraunitary vector filter bank theory and matrix theory. A novel method for constructing a kind of orthogonal shortly supported vector -valued wavelets is presented.

The Nice Feature of a Sort of Orthogonal Finitely Supported Five-Variant Wavelet Packages

2012 ◽  
Vol 459 ◽  
pp. 289-292
Author(s):  
Hong Wei Gao ◽  
Lan Ran Fang
Keyword(s):  
Iteration Method ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Wavelet Frames ◽  
New Approach ◽  
Decomposition Scheme ◽  
Necessary And Sufficient

In this article, the notion of orthogonal nonseparable five-variant wavelet packages, whi- ch is the generalization of orthogonal univariate wavelet packages, is introduced. A new approach for constructing the wavelet packages is presented by iteration method as well wavelet frames. The orthogonality properties for five-dimensional wavelet packages are discussed. Three orthogonality formulas concerning these wavelet packages are estabished. A necessary and sufficient condition for the existence of the pyramid decomposition scheme of space is presented.

scholarly journals Vector-valued nonuniform multiresolution analysis related to Walsh function

2020 ◽  
Vol 8 (1) ◽  
pp. 206-219
Author(s):  
Abdullah
Keyword(s):  
Multiresolution Analysis ◽  
Walsh Function ◽  
Half Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Vector Valued ◽  
Positive Half

In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh function. We obtain the necessary and sufficient condition for the existence of associated wavelets.

-NONUNIFORM MULTIRESOLUTION ANALYSIS

2021 ◽  
pp. 1-19
Author(s):  
S. PITCHAI MURUGAN ◽  
G. P. YOUVARAJ
Keyword(s):  
Signal Processing ◽  
Multiresolution Analysis ◽  
Sufficient Conditions ◽  
Discrete Subgroup ◽  
Scaling Functions ◽  
Wavelet Filters ◽  
Spectral Pairs ◽  
Dilation Factor ◽  
Necessary And Sufficient ◽  
Funct Anal

Abstract Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal.158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

WAVELETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF-LINE

2012 ◽  
Vol 10 (02) ◽  
pp. 1250018 ◽  
Author(s):  
MEENAKSHI ◽  
P. MANCHANDA ◽  
A. H. SIDDIQI
Keyword(s):  
Multiresolution Analysis ◽  
Scaling Function ◽  
Half Line ◽  
Necessary And Sufficient Condition ◽  
Locally Compact Abelian Groups ◽  
Spectral Pairs ◽  
Compact Abelian Groups ◽  
Necessary And Sufficient ◽  
The Scaling Function ◽  
Positive Half

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,3 has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L2(ℝ+). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V0 of L2(ℝ+) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}λ∈Λ+ where Λ+ = {0, r/N} + ℤ+, N > 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ+ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

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