Construction of Periodic Wavelet Frames Generated by the Walsh Polynomials

<p>In view of the good properties of nonstationary wavelet frames and the better flexibility of wavelets in Sobolev spaces, the nonstationary dual wavelet frames in a pair of dual Sobolev spaces are studied in this paper. We mainly give the oblique extension principle and the mixed extension principle for nonstationary dual wavelet frames in a pair of dual Sobolev spaces Hs(Rd) and H-s(Rd).</p>

Intuitionistic fuzzy sets (IFS) are more applicable and reliable to deal with the real life situations under uncertain environment. The extension principle is a vital concept in fuzzy field specially in fuzzy arithmetic. In this article, a pair of extension principles namely minimal extension principle and average extension principle are proposed for IFS with some associated properties. Moreover, some properties of general extension principle for IFSs are explored. Finally, arithmetic operations for IFSs based on the average extension principle are developed with examples.

Construction of nonseparable dual [formula omitted]-wavelet frames in [formula omitted

It is shown that there exists a frame wavelet with fast decay in the time domain and compact support in the frequency domain generating a wavelet system whose canonical dual frame cannot be generated by an arbitrary number of generators. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.

Asymptotic Property of Dual Frames of a Wavelet Frame Yielded by Several Refinable Functions

Wavelet analysis is a particular time-or space-scale representation of signals which has found a wide range of applications in physics, signal processing and applied Mathematics in the last few years. We show that there exist wavelet frame generated by two functions which have good dual wavelet frames, but for which the canonical dual wavelet frame does not consist of wavelets, according to scaling functions.

Hybrid systems, which consist of continuous-state and discrete-state parts are considered. It is assumed that the discrete-state part of the system is event-driven. Inclusion and extension principles are first defined for such systems. The controller design problem for such systems is then considered within the framework of the extension principle. It is shown that if the extension principle is used then any controller designed in the expanded spaces is contractible to the original spaces for implementation. Stability of hybrid systems is also considered. It is shown that if a system, which includes another system, is stable then the included system is also stable. Furthermore, if a controller designed for the expanded system achieves stability then the contracted controller also achieves stability for the original system. All the results may be specialized to the case of discrete-event dynamic systems.

Starting from any two compactly supported d-refinable function vectors in (L2(R))r with multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual d-wavelet frames in L2(R) and they achieve the best possible orders of vanishing moments. When all the components of the two real-valued d-refinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual d-wavelet frames, can be real-valued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any d-refinable function vector are also considered. This paper generalizes the work in [5,12,13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.

Density order of Parseval wavelet frames from extension principles

The unitary extension principle by Ron and Shen is one of the cornerstones of wavelet frame theory; it leads to tight frames for L 2 ( R ) L^{2}(\mathbb {R}) and associated expansions of functions f ∈ L 2 ( 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.

Preface Contributors Color Insert I. Wavelets and Wavelet Transforms Wavelet Frames: Multiresolution Analysis and Extension Principles J.J. Benedetto and O.M. Treiber Convergence Rates of Multiscale and Wavelet Expansions M.A. Kon and L.A. Raphael Denoising via Nonorthogonal Wavelet Transforms K. Berkner and R.O. Wells, Jr. Osiris Wavelets and the Dipole Gas G. Battle Wavelets in Closed Forms A. Zayed and G.G. Walter Wavelet Galerkin Methods for Boundary Integral Equations and the Coupling with Finite Element Methods C. Perez and R. Schneider Computing and Analyzing Turbulent Flows Using Wavelets K. Schneider and M. Farge The Uncertainty Principle for the Short-Time Fourier Transform and Wavelet Transform L. Cohen II. Time-Frequency Signal Analysis Quadratic Time-Frequency Analysis of Linear, Time-Varying Systems F. Hlawatsch and G. Matz Inequalities in Mellin--Fourier Signal Analysis P. Flandrin Introduction to Time-Frequency Signal Analysis B. Boashash and B. Barkat Reduced Interference Time-Frequency Distributions: Scaled Decompositions and Interpretations / W.J. Williams Index

Three new cut sets of fuzzy sets and new theories of fuzzy sets

This paper presents a new approach to dynamic system state estimation under bounded noises via the Evidential Reasoning(£K) rule. This method regards the dynamic system equations and the actual observations of the system states as two information sources. The random set description of evidence and the extension principle of random set are used to recursively generate state evidence and observation evidence respectively from the two information sources and to propagate them in system equations. At each time step, the ER rule is used to fuse the two pieces of evidence in observation domain and then the fused result is transformed to state domain by the extension principles. Pignistic expectation of the fused result is calculated as state estimation value. Compared with the estimation method using interval analysis and evidence theory given by Nassreddine, the proposed approach makes estimation results more accurate by using fusion mechanism of the ER rule considering weight and reliability of evidence. The method is shown to have better performances in an application to liquid level estimation of industrial level apparatus than does the Nassreddine's method.

In this paper, a Littlewood-Paley function characterization of the spaces L p (R), 1 < p< ∞ is first established by means of the equivalent conditions of tight wavelet frames, wherein the Littlewood-Paley function is associated with a tight wavelet frame generated by the so-called extension principles. With the above characterization, another characterization of L p (R), 1 < p< ∞ is also established in terms of the weighted l 2-norm of the wavelet frame coefficients, which can be a useful tool in harmonic analysis, approximation theory, and image processing.