-NONUNIFORM MULTIRESOLUTION ANALYSIS

Author(s):  
S. PITCHAI MURUGAN ◽  
G. P. YOUVARAJ

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.

Author(s):  
Xiaojiang Yu

Frame multiresolution analysis (FMRA) in [Formula: see text] is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in [Formula: see text], which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity [Formula: see text]. We characterize wavelet frames and Parseval wavelet frames for [Formula: see text] under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions [Formula: see text] in [Formula: see text] to be multiframe generators of [Formula: see text]. Especially, the second condition depends on the multiscaling FMRA and [Formula: see text] only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.


2018 ◽  
Vol 50 (2) ◽  
pp. 119-132
Author(s):  
Mohd Younus Bhat

A generalization of Mallats classical multiresolution analysis, based on thetheory of spectral pairs, was considered in two articles by Gabardo and Nashed. In thissetting, the associated translation set is no longer a discrete subgroup of R but a spectrumassociated with a certain one-dimensional spectral pair and the associated dilation is aneven positive integer related to the given spectral pair. In this paper, we construct dualwavelets which are associated with Nonuniform Multiresolution Analysis. We show thatif the translates of the scaling functions of two multiresolution analyses are biorthogonal,then the associated wavelet families are also biorthogonal. Under mild assumptions onthe scaling functions and the wavelets, we also show that the wavelets generate Rieszbases


2018 ◽  
Vol 10 (2) ◽  
pp. 340-346
Author(s):  
◽  
Firdous A. Shah

Abstract A generalization of Mallat’s classic theory of multiresolution analysis based on the theory of spectral pairs was considered by Gabardo and Nashed [4] for which the 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. In this short communication, we characterize the scaling functions associated with such a nonuniform multiresolution analysis by means of some fundamental equations in the Fourier domain.


Author(s):  
Hari Krishan Malhotra ◽  
Lalit Kumar Vashisht

The main purpose of this paper is to provide a characterization of scaling functions for non-uniform multiresolution analysis (NUMRA, in short). Some necessary and sufficient conditions for scaling functions of wavelet NUMRA in the frequency domain are also obtained.


Author(s):  
M. Younus Bhat

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.


2015 ◽  
Vol 08 (02) ◽  
pp. 1550034 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda

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.


2013 ◽  
Vol 06 (01) ◽  
pp. 1350007 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda

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 a group. Farkov [Orthogonal p-wavelets on ℝ+, in Proc. Int. Conf. Wavelets and Splines (St. Petersburg State University, St. Petersburg, 2005), pp. 4–26] studied multiresolution analysis (MRA) on positive half line and constructed associated wavelets. Meenakshi et al. [Wavelets associated with Nonuniform multiresolution analysis on positive half line, Int. J. Wavelets, Multiresolut. Inf. Process.10(2) (2011) 1250018, 27pp.] studied NUMRA on positive half line and proved the analogue of Cohen's condition for the NUMRA on positive half line. We construct the associated wavelet packets for such an MRA and study its properties.


2008 ◽  
Vol 21 (3) ◽  
pp. 309-325 ◽  
Author(s):  
Yury Farkov

This paper gives a review of multiresolution analysis and compactly sup- ported orthogonal wavelets on Vilenkin groups. The Strang-Fix condition, the partition of unity property, the linear independence, the stability, and the orthonormality of 'integer shifts' of the corresponding refinable functions are considered. Necessary and sufficient conditions are given for refinable functions to generate a multiresolution analysis in the L2-spaces on Vilenkin groups. Several examples are provided to illustrate these results. .


2021 ◽  
Vol 7 (1) ◽  
pp. 3
Author(s):  
Ishtaq Ahmed ◽  
Owias Ahmad ◽  
Neyaz Ahmad Sheikh

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.


2016 ◽  
Vol 18 (06) ◽  
pp. 1550072 ◽  
Author(s):  
El Hadji Abdoulaye Thiam

Let [Formula: see text] be a smooth compact Riemannian manifold of dimension [Formula: see text] and let [Formula: see text] to be a closed submanifold of dimension [Formula: see text]. In this paper, we study existence and non-existence of minimizers of Hardy inequality with weight function singular on [Formula: see text] within the framework of Brezis–Marcus–Shafrir [Extremal functions for Hardy’s inequality with weight, J. Funct. Anal. 171 (2000) 177–191]. In particular, we provide necessary and sufficient conditions for existence of minimizers.


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