Orthonormal Wavelet Bases Research Articles

Bayes synthesis of multidimensional stochastic system with high availability by wavelet canonical expansions

Paper in devoted to methodological and instrumental software synthesis support for multidimensional stochastic systems with high availability (StSHA). For modeling of nonstationary stochastic processes (StP) it is supposed to implement wavelet canonical expansions (WLCE), based on coefficients of canonical expansions of covariance matrix using two dimensional orthonormal wavelet basis with compact carrier. Vector StSHA input and output are nonlinear on vector of random parameters with known probability density and Gaussian (normal) noise. Noises do not depend upon random parameters. Functional Bayes criterion (FBC) is presented by linear functional. Optimal system is synthesized using conditional risk criterion. General StSHA synthesis method and algorithm contains 4 steps and 7 positions. FBC optimal estimate is calculated by WLCE method. Special software is developed for scalar nonlinear StSHA. Results of computer experiments are given and discussed. His shown that high efficiency of method at 8 WLCE items. Directions of future investigations are presented.

A new algorithm of boundary value problems based on improved wavelet basis and the reproducing kernel theory

Relying upon Legendre polynomials, an improved multiwavelet orthonormal basis is constructed in the reproducing kernel space. We study boundary value problems (BVPs) based on the improved wavelet orthonormal basis and construct the ‐best approximate solution. Convergence and the stability are discussed; this method has higher order convergence. The main advantage of the wavelet based our method is its efficiency and simple applicability for a variety of boundary conditions. Some numerical tests even including singular BVPs reveal the stability and efficiency of the new algorithm.

Existence of unconditional wavelet bases for Lp-norm over a local fields of positive characteristic

In this paper, we defined Calder[Formula: see text]n–Zygmund operator [Formula: see text] as [Formula: see text] on local fields ([Formula: see text]) under certain conditions on [Formula: see text] and then find the boundedness of operator [Formula: see text]. Using result on boundedness of [Formula: see text] and orthonormal wavelet basis of [Formula: see text], we provide unconditional wavelet bases for [Formula: see text]-norm over a local fields of positive characteristic by the wavelet coefficients.

Chebyshev wavelet-Picard technique for solving fractional nonlinear differential equations

Abstract In the present paper, an efficient method based on a new kind of Chebyshev wavelet together with Picard technique is developed for solving fractional nonlinear differential equations with initial and boundary conditions. The new orthonormal Chebyshev wavelet basis is constructed from a class of orthogonal polynomials called the fifth-kind Chebyshev polynomials. The convergence analysis and error estimation of the proposed Chebyshev wavelet expansion are studied. An exact formula for the Riemann-Liouville fractional integral of the Chebyshev wavelet is derived. Picard iteration is used to convert the fractional nonlinear differential equations into a fractional recurrence relation and then the proposed Chebyshev wavelet collocation method is applied on the converted problem. Several test problems are given to illustrate the performance and effectiveness of the proposed method and compared with the existing work in the literature.

Explicit commutative sequence space representations of function and distribution spaces on the real half-line

We provide explicit commutative sequence space representations for classical function and distribution spaces on the real half-line. This is done by evaluating at the Fourier transforms of the elements of an orthonormal wavelet basis.

A New Algorithm Based on Bernstein Polynomials Multiwavelets for the Solution of Differential Equations Governing AC Circuits

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

The unitary extension principle for locally compact abelian groups with co-compact subgroups

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames forL2(R)L^{2}(\mathbb {R})and associated expansions of functionsf∈L2(R)f\in L^{2}(\mathbb {R})of similar type as those for orthonormal wavelet bases. In this paper, the unitary extension principle is extended to the setting of a locally compact abelian group, equipped with a collection of nested co-compact subgroups. Unlike all previously known generalizations of the unitary extension principle, the current one is taking place within the setting of continuous frames, which means that the resulting decompositions of functions in the underlying Hilbert space in general are given in terms of integral representations rather than discrete sums. The frame elements themselves appear by letting a collection of modulation operators act on a countable family of basic functions.

A new class of fredholm integral equations of the second kind with non symmetric kernel: solving by wavelets method

In this paper, we present an ecient modication of the wavelets method to solve a new class of Fredholm integral equations of the second kind with non symmetric kernel. This -analytical method based on orthonormal wavelet basis, as a consequence three systems are obtained, a Toeplitz system and two systems with condition number close to 1. Since the preconditioned conjugate gradient normal equation residual (CGNR) and preconditioned conjugate gradient normal equation error (CGNE) methods are applicable, we can solve the systems in O(2n log(n)) operations, by using the fast wavelet transform and the fast Fourier transform.

Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure $$\mu $$ satisfies the doubling property only. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure $$\mu .$$ As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel–Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher–Hytonen on spaces of homogeneous type in the sense of Coifman and Weiss.

Orhonormal Wavelet Bases on The 3D Ball Via Volume Preserving Map from the Regular Octahedron

We construct a new volume preserving map from the unit ball B 3 to the regular 3D octahedron, both centered at the origin, and its inverse. This map will help us to construct refinable grids of the 3D ball, consisting in diameter bounded cells having the same volume. On this 3D uniform grid, we construct a multiresolution analysis and orthonormal wavelet bases of L 2 ( B 3 ) , consisting in piecewise constant functions with small local support.

Wavelets on compact abelian groups

Abstract Multiresolution analysis (MRA) on a compact abelian group G has been constructed with epimorphism as a dilation operator. We show a characterization of scaling sequences of an MRA on L p ( G ) , 1 ≤ p ∞ . With the help of the scaling sequence we construct an orthonormal wavelet basis of L 2 ( G ) .

Wavelet frames in $L^2(\mathbb{R}^d)$

We obtain a necessary and sufficient condition for the existence of wavelet frames. We define and study the synthesis and analysis operators associated with wavelet frames. We discuss some applications of operator value (OPV) frames in the theory of wavelet frames. Also, we discuss the minimal property of wavelet frame coefficients and study the property of over completeness of wavelet frames. Various characterizations of wavelet frame, Riesz wavelet basis and orthonormal wavelet basis are given. Further, dual wavelet frames are discussed and a characterization of dual wavelet frames is given. Finally, we give a characterization of a pair of biorthogonal Riesz bases.

Approximation of Functions by n-Separate Wavelets in the Spaces Lp(ℝ), 1 ≤ p ≤ ∞

We consider the orthonormal bases of n-separate MRAs and wavelets constructed by the author earlier. The classical wavelet basis of the space L2(ℝ) is formed by shifts and compressions of a single function ψ. In contrast to the classical case, we consider a basis of L2(ℝ) formed by shifts and compressions of n functions ψs, s = 1,...,n. The constructed n-separate wavelets form an orthonormal basis of L2(ℝ). In this case, the series $$\sum\nolimits_{s = 1}^n {\sum\nolimits_{j \in {\rm Z}} {\sum\nolimits_{k \in {\rm Z}} {f,\psi _{nj + s}^s >\psi _{nj + s}^s} } } $$ converges to the function f in the space L2(ℝ). We write additional constraints on the functions ϕs and ψs, s = 1,..., n, that provide the convergence of the series to the function f in the spaces Lp(ℝ), 1 ≤ p <- ∞, in the norm and almost everywhere.

New Calderón reproducing formulae with exponential decay on spaces of homogeneous type

Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytonen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderon reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderon reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

Characteristic Vectors in p/q-Channel Orthonormal Wavelet

Wavelet analysis is a newly rapidly developing subject in the late twentieth century. As a time-frequency analysis tool, wavelet analysis has many advantages over other time-frequency tools, such as in signal processing, image processing, speech processing, pattern recognition, quantum physics and other fields. Multiresolution analysis (MRA for short) is an important method for studying wavelet orthonormal wavelet bases with rational dilation 2. However, p/q-band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing and attracted more and more interest in recent years. But there are relatively less results for the case of p/q-band. This paper studies the orthonormal wavelet bases with rational dilation factor p/q based on multiresolution analysis by a polyphase decomposition technique. First, we gave the concept of Multiresolution analysis with rational dilation p/q and deduced an identity of the masks matrix. Also, a perfect reconstruction condition in terms of masks was presented. Further, we gave the refinement and wavelet matrices respectively and derived the characteristic roots and the corresponding orthonormal characteristic vectors of the wavelet matrix, and then a method with characteristic vectors was reduced to achieve the orthonormal wavelet bases with rational dilation factor p/q. In the end, an example is offered to verify this theory.

A direct numerical solution of time-delay fractional optimal control problems by using Chelyshkov wavelets

The aim of the present study is to present a numerical algorithm for solving time-delay fractional optimal control problems (TDFOCPs). First, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet functions and their properties are implemented to derive some operational matrices. Then, the fractional derivative of the state function in the dynamic constraint of TDFOCPs is approximated by means of the Chelyshkov wavelets. The operational matrix of fractional integration together with the dynamical constraints is used to approximate the control function directly as a function of the state function. Finally, these approximations were put in the performance index and necessary conditions for optimality transform the under consideration TDFOCPs into an algebraic system. Moreover, some illustrative examples are considered and the obtained numerical results were compared with those previously published in the literature.

Precise Modeling of Reverberant Room Responses Using Wavelet Decomposition and Orthonormal Basis Functions

Precise Modeling of Reverberant Room Responses Using Wavelet Decomposition and Orthonormal Basis Functions

Numerical solution of systems of fractional delay differential equations using a new kind of wavelet basis

In the present paper, a new orthonormal wavelet basis, called Chelyshkov wavelet, is constructed from a class of orthonormal polynomials. These wavelet basis and their properties are utilized to obtain their operational matrix of fractional integration in the Riemann–Liouville sense and delay operational matrix. Convergence and error bound of the expansion by this kind of wavelet functions are investigated. Then, these operational matrices along with the Galerkin approach have been implemented to solve systems of fractional delay differential equations (SFDDEs). The main superiority of the proposed technique is that it reduces SFDDEs to a system of algebraic equations. Moreover, accuracy and efficiency of the suggested Chelyshkov wavelet approach are verified through some linear and nonlinear SFDDEs. Finally, the obtained numerical results are compared with those previously reported in the literature.

Unconditional wavelet bases in Lebesgue spaces

In this paper, we prove that an orthonormal wavelet basis associated with a general isotropic expansive matrix must be an unconditional basis for all $L^{p}$(ℝ${}^{d})$ with $1$<$p$<$\infty$, provided the wavelet functions satisfy some usual conditions.

An Explicit Algorithm for the Construction of 3-Band Wavelet Frames Based on FMRA

In 1974, an French engineer, J. Morlet raised the concept of wavelet transform and established the inversion formula through the experience of physical intuition and signal processing and in 1986, the famous mathematician, Y. Meyer, created a real small wave base. From then on, Wavelet transform is a rapidly developing new subfield in mathematics and is used in more and more fields, such as signal analysis, image processing, quantum mechanics and theoretical physics etc. Multiresolution analysis is a systematic method for constructing orthonormal wavelet bases and most of the current dilation is M=2 With the development of wavelet transform, M>2 -band wavelet is known to have advantages over 2-band wavelet in some aspects such as in signal processing. However, there are relatively less results for the case of M> 2. Based on this fact and inspired by other similar papers, this paper studies the 3-band wavelets and wavelet frames associated with a given refinable function based on frame multiresolution analysis. In this paper, firstly, a sufficient and necessary condition which the refinable function should satisfy for the existence of wavelet frames is showed. Further, an explicit algorithm to construct this frames is worked out and finally, several designed examples are constructed to illustrate this algorithm.