Orthogonality Formulas Research Articles

Construction of vector valued wavelet packets on ℝ+ using Walsh-Fourier transform

In this paper, the concept of vector-valued wavelet packets in space L 2(ℝ+, ℂ N ) is introduced. Some properties of vector-valued wavelets packets are studied and orthogonality formulas of these wavelets packets are obtained. New orthonormal basis of L 2(ℝ+, ℂ N ) is obtained by constructing a series of subspaces of vector-valued wavelet packets.

The Characteristics of Trivariate Vector-Valued Super-Wave Wraps and Applications in Materials Engineering

The rise of wavelet analysis in applied mathematics is due to its applications and the flex- ibility. We introduce vector-valued wave wraps with multi-scale dilation factor for space , which is the generalization of univariate wavelet packets. A method for constructing a sort of orthogo-nal vector-valued wavelet wraps in three-dimensional space is presented and their orthogonality traits are characterized by virtue of iteration method and time-frequency analysis method. The orthog-onality formulas concerning these wave packets are established. Moreover, it is shown how to obtain new Riesz bases of space from these wave wraps. The notion of multiple pseudo fames for subspaces with integer translation is proposed. The construction of a generalized multiresolution structure of Paley-Wiener subspaces of is investigated. The pyramid decomposition scheme is derived based on a generalized multiresolution structure.

The Study of Compactly Supported Orthogonal Two-Direction Vector-Valued Wavelet Packet

The concept of two-direction vector-valued wavelet packets is proposed. An approach for constructing orthogonal two-direction vector-valued wavelet packets is developed and their properties are discussed by means of time-frequency analysis method, algebra theory and functional analysis method. Orthogonality formulas concerning these two-direction vector-valued wavelet packets are established.

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

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.

A Study of Vector-Valued Binary Scaling Functions and Parseval Frames with Integer Dilation Constant

Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. It is the branch of engineering that involves the production and usage of heat and mechanical power for the design, production, and operation of machines and tools. It is one of the oldest and broadest engineering disciplines. The notion of orthogonal vector-valued binary wavelet packs is introduced. Their traits is investigated by virtue of time-frequency analysis method and finite group theory. Orthogonality formulas are established. Orthonormal wavelet pack bases are obtained. A novel method for constructing a kind of orthogonal shortly supported vector-valued wavelets is presented.

The Characteristics Concerning a Class of Orthogonal Multivariate Matrix-Valued Scaling Functions and Frame Packets

Mechanical engineering is a discipline of engineering that applies the principles of physics and materials science for analysis, design, manufacturing, and maintenance of mechanical systems. In this work, the notion of matrix-valued multiresolution analysis of space is introduced. A method for constructing orthogonal matrix-valued ternary wavelet packs is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three orthogonality formulas concerning these wavelet packets are provided. Finally, new orthonormal wavelet pack bases of space are obtained by constructing a series of subspaces of orthogonal matrix-valued wavelet packets.

The Presentation of Biorthogonal Non-Tensor Trivariate Wavelet Wraps in Besov Space

In this paper, the concept of orthogonal non-tensor bivariate wavelet packs, which is the generalization of orthogonal univariate wavelet packs, is pro -posed by virtue of analogy method and iteration method. Their orthogonality property is investigated by using time-frequency analysis method and variable se-paration approach. Three orthogonality formulas regarding these wavelet wraps are established. Moreover, it is shown how to draw new orthonormal bases of space from these wavelet wraps. A procedure for designing a class of orthogonal vector-valued finitely supported wavelet functions is proposed by virtue of filter bank theory and matrix theory.

The Features of a Class of Orthogonal Multiple Vector Ternary Wavelet Wraps and Applications in Physics

Wavelet analysishas been a powerful tool for exploring and solving many complicated problems in natural science and enginee-ring computation. In this work, the notion of orthogonal matrix-valued trivariate small-wave wraps and wavelet frame wraps, which are generalization of univariate small-wave wraps, is introduced. A new procedure for constructing these vector trivariate small-wave wraps is presented. Their characteristics are studied by using time-frequency analysis method, Banach space theory and finite group theory. Orthogonal formulas concerning the wavelet packs are established. The biorthogonality formulas concerning these wavelet wraps are established. Moreover, it is shown how to draw new Riesz bases of space from these wavelet wraps.

The Traits of Nontensor Five-Variant Wavelet Packs and Applications in Management Science and Materials Science

Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this article, the notion of orthogonal nonseparable five-variant wavelet packages is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A feasible approach for constructing two-directional orthogonal wavelet packs is developed. The orthogonality property of five-variant wavelet packs is discussed. Three orthogonality formulas concerning these wavelet packs are estabished. A constructive method for affine frames of is proposed. The sufficient condition for the existence of a class of affine pseudoframes with filter banks is obtained by virtue of a generalized multiresolution analysis.

The Traits of Orthogonal Quarternary Small Function Wraps according to Quantity Matrix Dilation

The advantages of wavelet packets and their promising featu-res in various application have attracted a lot of interest and effort in recent years. In this paper, we propose the notion of quarternary small function wraps according to a quantity matrix dilation, which are gener-alization of univariate wavelet wraps. A nice procedure for designing the vector-valued quarternary small-wave wraps is provided. Their cha-racteristics are researched by virtue of time-frequency analysis method, wavelet transform and curvelet coefficients. The orthogonality formulas concerning these small-wave wraps are established. Furthermore, it is shown how to draw new orthogonal bases of space from the small-wave wraps. A method for designing a class of affine quarternary dual frames in four-dimensional space is presented. The results we obtain gains much improvement

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

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.

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

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.

The Traits of Two-Directional Orthogonal Multivariate Small-Wave Packages with Finite Support

In this paper, we introduce the notion of vector-valued multiresolution analysis and two-directional vector-valued multivariate wavelet packages associated with an integer-valued dilation matrix. A novel method for constructing multi-dimensional vector-valued wavelet packet is presented. Their characteristics are researched by means of operator theory, time-frequency analysis method and matrix theory. Three orthogonality formulas concerning the wavelet packets are establ- lished. Orthogonality decomposition relation formulas of the space are derived by constru- cting a series of subspaces of wavelet packets.

The Orthogonality Characters of Vector Binary Function Packets with an Integer Dilation Factor

Abstract Wavelet analysis is nowadays a widely used tool in applied mathematics. In this work the concept of orthogonal vector trivariate wavelet packets, which is a generalization of uni-wavelet packets, is introduced. A new method for constructing them is developed, and their characteristics is investigated by using time-frequency analysis method, matrix theory and iteration method. Orthogonality formulas are established. The sufficient condition for the existence of is investigated. Moreover, it is shown how to gain new orthonormal bases of space L 2 ( R 2 ) r from these wavelet packets.

The Optimality Characters of Vector-Valued Binary Small-Wave Wraps with a Constant Dilation Factor

Frame theory has been the focus of active research for twenty years, both in theory and applications. In this work, the notion of orthogonal vector-valued binary small-wave wraps, which is a generalization of uni-wavelet packets, is introduced. A procedure for constructing them is presented. Their orthogonality traits are investigated by using time-frequency analysis method, matrix theory and finite group theory. Orthogonality formulas concerning these binary small-wave wraps are established. Orthonormal wavelet packet bases are obtained. The pyramid decomposition scheme is obtained based on such a GMS and a sufficient condition for its existence is provided.

The Orthogonality Characters of Multiple Vector-Valued Ternary Wavelets and Applications in Theoretical Physics

Wavelet analysis has been the focus of active research for twenty years, both in theory and applications. In this work, we develop the concept of a class of multiple vector-valued trivariate wavelet wraps with a dilation matrix. A new method for constructing multiple vector-valued trivariate wavelet wraps is proposed. Their characters are investigated by means of operator technique, time-frequency analysis method and matrix theory. There orthogonality formulas regarding the wavelet wraps are provided. Orthogonality decomposition relation formulas of the space L2(R3, Cr×r)are obtained by constructing a series of subspaces of the multiple vector-valued wavelet wraps. Furthermore, several orthonormal wavelet wrap bases of space L2(R3, Cr×r) are constructed from the wavelet wraps. The pyramid decomposition scheme is obtained based on such a GMS and a sufficient condition for its existence is provided. Relation to theoretical physics is also discussed.

The Excellent Traits of a Class of Orthogonal Quarternary Wavelet Wraps with Short Support

The frame theory has been one of powerful tools for researching into wavelets. In this article, the notion of orthogonal nonseparable quarternary wavelet wraps, which is the generalizati- -on of orthogonal univariate wavelet wraps, is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A liable approach for constructing two-directional orthogonal wavelet wraps is developed. The orthogonality property of quarternary wavelet wraps is discussed. Three orthogonality formulas concerning these wavelet wraps are estabished. A constructive method for affine frames of L2(R4) is proposed. The sufficient condition for the existence of a class of affine pseudoframes with filter banks is obtained by virtue of a generalized multiresolution analysis. The pyramid decomposition scheme is established based on such a generalized multiresolution structure.

A Study of Orthogonal Vector-Valued Binary Wavelets and Wavelet Packs with Integer Dilation Constant

The notion of orthogonal vector-valued binary wavelet packs is introduced. Their traits is investigated by virtue of time-frequency analysis method and finite group theory. Orthogonality formulas are established. Orthonormal wavelet pack bases are obtained. A novel method for constructing a kind of orthogonal shortly supported vector-valued wavelets is presented.

An Algorithm and Characteristics Concerning a Sort of Orthogonal Ternary Matrix-Valued Frame Wavelet Packs

In this work, the notion of matrix-valued multiresolution analysis of space L2(Rn, Cs×s) is introduced. A method for constructing orthogonal matrix-valued ternary wavelet packs is developed and their properties are discussed by means of time-frequency analysis method, matrix theory and functional analysis method. Three orthogonality formulas concerning these wavelet packets are provided. Finally, new orthonormal wavelet pack bases of space L2(Rn, Cs×s) are obtained by constructing a series of subspaces of orthogonal matrix-valued wavelet packets.

The Research of Exact Orthogonal Nonseparable Four-Variate Wavelet Pack Bases with Finite Support

Wavelet analysis has become a popular subject in scientific research for twenty years. It has been a powerful tool for exploring and solving many complica -ted problems in natural science and engineering computation. The notion of exact orthogonal nonseparable four-variate wavelet packs is introduced. A novel method for constructing them is provided by iteration method. The orthogonality properties of four-variate wavelet packs are discussed. Three orthogonality formulas concerning these wavelet packs are provided. The pyramid decomposition scheme is derived