scholarly journals Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
Owais Ahmad ◽  
Neyaz A. Sheikh ◽  
Mobin Ahmad
Keyword(s):  
Multiresolution Analysis ◽  
Linear Span ◽  
Complete Characterization ◽  
Riesz Bases ◽  
Half Line ◽  
Scaling Functions ◽  
Biorthogonal Wavelet ◽  
Biorthogonal Wavelets ◽  
Single Function ◽  
Positive Half

AbstractIn this article, we introduce the notion of nonuniform biorthogonal wavelets on positive half line. We first establish the characterizations for the translates of a single function to form the Riesz bases for their closed linear span. We provide the complete characterization for the biorthogonality of the translates of scaling functions of two nonuniform multiresolution analysis and the associated biorthogonal wavelet families in $$L^2({\mathbb {R}}^+)$$ L 2 ( R + ) . Furthermore, under the mild assumptions on the scaling functions and the corresponding wavelets associated with nonuniform multiresolution analysis, we show that the wavelets can generate Reisz bases.

WAVELET PACKETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF LINE

2013 ◽  
Vol 06 (01) ◽  
pp. 1350007 ◽  
Author(s):  
Vikram Sharma ◽  
P. Manchanda
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packets ◽  
Half Line ◽  
State University ◽  
Spectral Pairs ◽  
Funct Anal ◽  
Positive Half

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.

scholarly journals Construction of a family of non-stationary biorthogonal wavelets

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Baoxing Zhang ◽  
Hongchan Zheng ◽  
Jie Zhou ◽  
Lulu Pan
Keyword(s):  
Multiresolution Analysis ◽  
Spline Functions ◽  
Refinable Functions ◽  
Biorthogonal Wavelet ◽  
Biorthogonal Wavelets ◽  
Vanishing Moments ◽  
B Spline ◽  
The Family ◽  
Cardinal Functions ◽  
The Stability

Abstract The family of exponential pseudo-splines is the non-stationary counterpart of the pseudo-splines and includes the exponential B-spline functions as special members. Among the family of the exponential pseudo-splines, there also exists the subclass consisting of interpolatory cardinal functions, which can be obtained as the limits of the exponentials reproducing subdivision. In this paper, we mainly focus on this subclass of exponential pseudo-splines and propose their dual refinable functions with explicit form of symbols. Based on this result, we obtain the corresponding biorthogonal wavelets using the non-stationary Multiresolution Analysis (MRA). We verify the stability of the refinable and wavelet functions and show that both of them have exponential vanishing moments, a generalization of the usual vanishing moments. Thus, these refinable and wavelet functions can form a non-stationary generalization of the Coifman biorthogonal wavelet systems constructed using the masks of the D–D interpolatory subdivision.

A Study of a Two-Directional Vector Multivariate Wavelet Wraps and Applications in Engineering Materials

2014 ◽  
Vol 889-890 ◽  
pp. 1270-1274
Author(s):  
Jin Shun Feng ◽  
Qing Jiang Chen
Keyword(s):  
Multiresolution Analysis ◽  
Riesz Bases ◽  
Scaling Functions ◽  
Sufficient Condition ◽  
Analysis Method ◽  
Compactly Supported ◽  
Dilation Matrix ◽  
Engineering Materials ◽  
Vector Valued

The existence of compactly supported orthogonal two-directional vector-valued wavelets associated with a pair of orthogonal shortly supported vector-valued scaling functions is researched. We introduce a class of two-directional vector-valued four-dimensional wavelet wraps according to a dilation matrix, which are generalizations of univariate wavelet wraps. Three orthogonality formulas regarding the wavelet wraps are established. Finally, it is shown how to draw new Riesz bases of space from these wavelet wraps. The sufficient condition for the existence of four-dimensional wavelet wraps is established based on the multiresolution analysis method.

ON BIORTHOGONAL WAVELETS RELATED TO THE WALSH FUNCTIONS

2011 ◽  
Vol 09 (03) ◽  
pp. 485-499 ◽  
Author(s):  
YU. A. FARKOV ◽  
A. YU. MAKSIMOV ◽  
S. A. STROGANOV
Keyword(s):  
Numerical Experiments ◽  
Half Line ◽  
Walsh Functions ◽  
Similar Technique ◽  
Biorthogonal Wavelets ◽  
Vilenkin Groups ◽  
Compactly Supported ◽  
Positive Half

In this paper, we describe an algorithm for computing biorthogonal compactly supported dyadic wavelets related to the Walsh functions on the positive half-line ℝ+. It is noted that a similar technique can be applied in very general situations, e.g., in the case of Cantor and Vilenkin groups. Using the feedback-based approach, some numerical experiments comparing orthogonal and biorthogonal dyadic wavelets with the Haar, Daubechies, and biorthogonal 9/7 wavelets are prepared.

scholarly journals Gabor systems on positive half line via Walsh-Fourier transform

2020 ◽  
Vol 12 (2) ◽  
pp. 468-482
Author(s):  
O. Ahmad ◽  
N.A. Sheikh
Keyword(s):  
Fourier Transform ◽  
Applied Mathematics ◽  
Vital Role ◽  
Complete Characterization ◽  
Half Line ◽  
Tight Frames ◽  
Gabor Systems ◽  
Orthonormal Bases ◽  
Positive Half

Gabor systems play a vital role not only in modern harmonic analysis but also in several fields of applied mathematics, for instances, detection of chirps, or image processing. In this paper, we investigate Gabor systems on positive half line via Walsh-Fourier transform. We provide the complete characterization of orthogonal Gabor systems on positive half line. Furthermore, we provide the characterization of tight frames and orthonormal bases of Gabor systems on positive half line in Fourier domain.

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.

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.

CONSTRUCTING MULTIPLE WAVELET PACKETS WITH FIFS

Fractals ◽  
2001 ◽  
Vol 09 (02) ◽  
pp. 165-169
Author(s):  
GANG CHEN ◽  
ZHIGANG FENG
Keyword(s):  
Multiresolution Analysis ◽  
Wavelet Packet ◽  
Wavelet Packets ◽  
Scaling Functions ◽  
Wavelet Packet Decomposition ◽  
Fractal Interpolation ◽  
Fractal Interpolation Functions ◽  
Interpolation Functions

By using fractal interpolation functions (FIF), a family of multiple wavelet packets is constructed in this paper. The first part of the paper deals with the equidistant fractal interpolation on interval [0, 1]; next, the proof that scaling functions ϕ1, ϕ2,…,ϕr constructed with FIF can generate a multiresolution analysis of L2(R) is shown; finally, the direct wavelet and wavelet packet decomposition in L2(R) are given.

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