ON BIORTHOGONAL WAVELETS RELATED TO THE WALSH FUNCTIONS

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.

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Nonuniform biorthogonal wavelets on positive half line via Walsh Fourier transform

Journal of the Egyptian Mathematical Society ◽

10.1186/s42787-021-00128-5 ◽

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.

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WAVELET PACKETS ASSOCIATED WITH NONUNIFORM MULTIRESOLUTION ANALYSIS ON POSITIVE HALF LINE

Asian-European Journal of Mathematics ◽

10.1142/s1793557113500071 ◽

2013 ◽

Vol 06 (01) ◽

pp. 1350007 ◽

Cited By ~ 2

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.

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Solvability of a Certain System of Singular Integral Equations with Convex Nonlinearity on the Positive Half-Line

Russian Mathematics ◽

10.3103/s1066369x21010035 ◽

2021 ◽

Vol 65 (1) ◽

pp. 27-46

Author(s):

Kh. A. Khachatryan ◽

H. S. Petrosyan

Keyword(s):

Integral Equations ◽

Singular Integral ◽

Singular Integral Equations ◽

Half Line ◽

Positive Half

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Asymptotic solution of Sturm-Liouville problem with periodic boundary conditions for relativistic finite-difference Schrödinger equation

Discrete and Continuous Models and Applied Computational Science ◽

10.22363/2658-4670-2020-28-3-230-251 ◽

2020 ◽

Vol 28 (3) ◽

pp. 230-251

Author(s):

Ilkizar V. Amirkhanov ◽

Irina S. Kolosova ◽

Sergey A. Vasilyev

Keyword(s):

Finite Difference ◽

Boundary Conditions ◽

Periodic Boundary ◽

Asymptotic Series ◽

Periodic Boundary Conditions ◽

Singularly Perturbed ◽

Half Line ◽

Asymptotic Convergence ◽

Relativistic Particles ◽

Positive Half

The quasi-potential approach is very famous in modern relativistic particles physics. This approach is based on the so-called covariant single-time formulation of quantum field theory in which the dynamics of fields and particles is described on a space-like three-dimensional hypersurface in the Minkowski space. Special attention in this approach is paid to methods for constructing various quasi-potentials. The quasipotentials allow to describe the characteristics of relativistic particles interactions in quark models such as amplitudes of hadron elastic scatterings, mass spectra, widths of meson decays and cross sections of deep inelastic scatterings of leptons on hadrons. In this paper SturmLiouville problems with periodic boundary conditions on a segment and a positive half-line for the 2m-order truncated relativistic finite-difference Schrdinger equation (LogunovTavkhelidzeKadyshevsky equation, LTKT-equation) with a small parameter are considered. A method for constructing of asymptotic eigenfunctions and eigenvalues in the form of asymptotic series for singularly perturbed SturmLiouville problems with periodic boundary conditions is proposed. It is assumed that eigenfunctions have regular and boundary-layer components. This method is a generalization of asymptotic methods that were proposed in the works of A. N. Tikhonov, A. B. Vasilyeva, and V. F Butuzov. We present proof of theorems that can be used to evaluate the asymptotic convergence for singularly perturbed problems solutions to solutions of degenerate problems when 0 and the asymptotic convergence of truncation equation solutions in the case m. In addition, the SturmLiouville problem on the positive half-line with a periodic boundary conditions for the quantum harmonic oscillator is considered. Eigenfunctions and eigenvalues are constructed for this problem as asymptotic solutions for 4-order LTKT-equation.

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