Charles K. Chui - Böcker

An Introduction to Wavelets is the first volume in a new series, WAVELET ANALYSIS AND ITS APPLICATIONS. This is an introductory treatise on wavelet analysis, with an emphasis on spline wavelets and time-frequency analysis. Among the basic topics covered in this book are time-frequency localization, integral wavelet transforms, dyadic wavelets, frames, spline-wavelets, orthonormal wavelet bases, and wavelet packets. In addition, the author presents a unified treatment of nonorthogonal, semiorthogonal, and orthogonal wavelets. This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis. It is suitable as a textbook for a beginning course on wavelet analysis and is directed toward both mathematicians and engineers who wish to learn about the subject. Specialists may use this volume as a valuable supplementary reading to the vast literature that has already emerged in this field. This is an introductory treatise on wavelet analysis, with an emphasis on spline-wavelets and time-frequency analysis This monograph is self-contained, the only prerequisite being a basic knowledge of function theory and real analysis Suitable as a textbook for a beginning course on wavelet analysis

This volume contains surveys written by leading experts, along with research papers on the developments in approximation theory and in the theory and application of curves and surfaces. Topics addressed include: approximation of eigenvalues, approximation with weights, Bernstein inequalities, Chebyshev centres, generalized polynomials, Harr sets, kernels on manifolds, linear operators, Markov inequalities, minimal projections, multivariate interpolation, nonlinear approximation, one-sided approximation, orthogonal polynomials, polynomial approximation, rational approximation, shift invariant spaces, uni-modular polynomials and zero-increasing transformations.

This volume contains surveys written by leading experts, along with research papers on the developments in approximation theory and in the theory and application of curves and surfaces. Topics addressed include: biharmonic splines, cardinal L-splines, discrete splines, exterior BVPs, financial optimization, framelets, FSI generators, Gabor frames, moving least squares, multiwavelets, nonstationary wavelets, non-MSF wavelets, orthogonal wavelets, penalized least squares, refinable functions, SAR interferometry, smoothing splines, spline bases, spline interpolation, Sturm-Liouville problems, subdivision, thin plate splines and wavelet packets.

The subject of multivariate splines has become a rapidly growing field of mathematical research. The author presents the subject from an elementary point of view that parallels the theory and development of univariate spline analysis. To compensate for the missing proofs and details, an extensive bibliography has been included. There is a presentation of open problems with an emphasis on the theory and applications to computer aided design, data analysis, and surface fitting. Applied mathematicians and engineers working in the areas of curve fitting, finite element methods, computer aided geometric design, signal processing, mathematical modelling, computer aided design, computer aided manufacturing, and circuits and systems will find this monograph essential to their research.

Wavelets continue to be powerful mathematical tools that can be used to solve problems for which the Fourier (spectral) method does not perform well or cannot handle. This book is for engineers, applied mathematicians, and other scientists who want to learn about using wavelets to analyze, process, and synthesize images and signals. Applications are described in detail and there are step-by-step instructions about how to construct and apply wavelets. The only mathematically rigorous monograph written by a mathematician specifically for nonspecialists, it describes the basic concepts of these mathematical techniques, outlines the procedures for using them, compares the performance of various approaches, and provides information for problem solving, putting the reader at the forefront of current research.New methods and important topics described in the book include the following:Multidimensional and multifrequency wavelet transforms (including wavelet packets). This approach gives the user more flexibility in applying wavelets to study multidimensional data sets.Spline wavelets with uniform or arbitrary knots on a bounded interval (with methods to construct these wavelets, their duals, as well as all decomposition and reconstruction matrices). This tool allows analysis and synthesis of discrete data on uniform or nonuniform sample sites without any boundary effect.Wavelets as a mathematical tool for waveform matching, signal segmentation, and time-frequency localization as well as effective implementation and fast computation.Procedures to construct all the well-known wavelets and to find their corresponding filter sequences.Detailed comparisons of the most popular wavelets and tables of values for evaluating their filtering performance.

Other topics include Kalman filtering for systems with correlated noise or colored noise, limiting Kalman filtering for time-invariant systems, extended Kalman filtering for nonlinear systems, interval Kalman filtering for uncertain systems, and wavelet Kalman filtering for multiresolution analysis of random signals.

Other topics include Kalman filtering for systems with correlated noise or colored noise, limiting Kalman filtering for time-invariant systems, extended Kalman filtering for nonlinear systems, interval Kalman filtering for uncertain systems, and wavelet Kalman filtering for multiresolution analysis of random signals.

Discrete H Optimization is concerned with the study of H optimization for digital signal processing and discrete-time control systems. The first three chapters present the basic theory and standard methods in digital filtering and systems from the frequency-domain approach, followed by a discussion of the general theory of approximation in Hardy spaces. AAK theory is introduced, first for finite-rank operators and then more generally, before being extended to the multi-input/multi-output setting. This mathematically rigorous book is self-contained and suitable for self-study. The advanced mathematical results derived here are applicable to digital control systems and digital filtering.

"Kalman Filtering with Real-Time Applications" presents a thorough discussion of the mathematical theory and computational schemes of Kalman filtering. The filtering algorithms are derived via different approaches, including a direct method consisting of a series of elementary steps, and an indirect method based on innovation projection. Other topics include Kalman filtering for systems with correlated noise or colored noise, limiting Kalman filtering for time-invariant systems, extended Kalman filtering for nonlinear systems, interval Kalman filtering for uncertain systems, and wavelet Kalman filtering for multiresolution analysis of random signals. The last two topics are new additions to this third edition. Most filtering algorithms are illustrated by using simplified radar tracking examples. The style of the book is informal, and the mathematics is elementary but rigorous. The text is self-contained, suitable for self-study, and accessible to all readers with a minimum knowledge.

A knowledge of linear systems provides a firm foundation for the study of optimal control theory and many areas of system theory and signal processing. State-space techniques developed since the early sixties have been proved to be very effective. The main objective of this book is to present a brief and somewhat complete investigation on the theory of linear systems, with emphasis on these techniques, in both continuous-time and discrete-time settings, and to demonstrate an application to the study of elementary (linear and nonlinear) optimal control theory. An essential feature of the state-space approach is that both time-varying and time-invariant systems are treated systematically. When time-varying systems are considered, another important subject that depends very much on the state-space formulation is perhaps real-time filtering, prediction, and smoothing via the Kalman filter. This subject is treated in our monograph entitled "Kalman Filtering with Real-Time Applications" published in this Springer Series in Information Sciences (Volume 17). For time-invariant systems, the recent frequency domain approaches using the techniques of Adamjan, Arov, and Krein (also known as AAK), balanced realization, and oo H theory via Nevanlinna-Pick interpolation seem very promising, and this will be studied in our forthcoming monograph entitled "Mathematical Ap­ proach to Signal Processing and System Theory". The present elementary treatise on linear system theory should provide enough engineering and mathe­ of these two subjects.

"Signal Processing and Systems Theory" is concerned with thestudy of H-optimization for digital signal processing anddiscrete-time control systems. The first three chapterspresent the basic theory and standard methods in digitalfiltering and systems from the frequency-domain approach,followed by a discussion of the general theory ofapproximation in Hardy spaces. AAK theory is introduced,first for finite-rank operators and then more generally,before being extended to the multi-input/multi-outputsetting. This mathematically rigorous book is self-containedand suitable for self-study. The advanced mathematicalresults derived here are applicable to digital controlsystems and digital filtering.

This textbook, apart from introducing the basic aspects of applied mathematics, focuses on recent topics such as information data manipulation, information coding, data approximation, data dimensionality reduction, data compression, time-frequency and time scale bases, image manipulation, and image noise removal.The methods treated in more detail include spectral representation and “frequency” of the data, providing valuable information for, e.g. data compression and noise removal. Furthermore, a special emphasis is also put on the concept of “wavelets” in connection with the “multi-scale” structure of data-sets.The presentation of the book is elementary and easily accessible, requiring only some knowledge of elementary linear algebra and calculus. All important concepts are illustrated with examples, and each section contains between 10 an 25 exercises. A teaching guide, depending on the level and discipline of instructions is included for classroom teaching and self-study.

This is the collection of the refereed and edited papers presented at the 8th Texas International Conference on Approximation Theory. It is interdisciplinary in nature and consists of two volumes. The central theme of Vol. I is the core of approximation theory. It includes such important areas as qualitative approximations, interpolation theory, rational approximations, radial-basis functions, and splines. The second volume focuses on topics related to wavelet analysis, including multiresolution and multi-level approximation, subdivision schemes in CAGD, and applications.