1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

Preface to the second edition Preface to the first edition 1. Background 2. Orthogonal polynomials in two variables 3. General properties of orthogonal polynomials in several variables 4. Orthogonal polynomials on the unit sphere 5. Examples of orthogonal polynomials in several variables 6. Root systems and Coxeter groups 7. Spherical harmonics associated with reflection groups 8. Generalized classical orthogonal polynomials 9. Summability of orthogonal expansions 10. Orthogonal polynomials associated with symmetric groups 11. Orthogonal polynomials associated with octahedral groups and applications References Author index Symbol index Subject index.

/pdf/orthogonal-polynomials-of-several-variables-2cwkynqk1v.pdf

Orthogonal Polynomials of Several Variables

This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift-invariant spaces. It is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift-invariant subspaces by bringing together wavelet theory, frame theory, reproducing kernel Hilbert spaces, approximation theory, amalgam spaces, and sampling. Inspired by applications taken from communication, astronomy, and medicine, the following aspects will be emphasized: (a) The sampling problem is well defined within the setting of shift-invariant spaces. (b) The general theory works in arbitrary dimension and for a broad class of generators. (c) The reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms. These algorithms converge geometrically and are robust in the presence of noise. (d) To model the natural decay conditions of real signals and images, the sampling theory is developed in weighted L p-spaces.

/pdf/nonuniform-sampling-and-reconstruction-in-shift-invariant-3fggp0zfbi.pdf

Nonuniform Sampling and Reconstruction in Shift-Invariant Spaces

Demand Response Management (DRM) is a key component in the smart grid to effectively reduce power generation costs and user bills. However, it has been an open issue to address the DRM problem in a network of multiple utility companies and consumers where every entity is concerned about maximizing its own benefit. In this paper, we propose a Stackelberg game between utility companies and end-users to maximize the revenue of each utility company and the payoff of each user. We derive analytical results for the Stackelberg equilibrium of the game and prove that a unique solution exists. We develop a distributed algorithm which converges to the equilibrium with only local information available for both utility companies and end-users. Though DRM helps to facilitate the reliability of power supply, the smart grid can be succeptible to privacy and security issues because of communication links between the utility companies and the consumers. We study the impact of an attacker who can manipulate the price information from the utility companies. We also propose a scheme based on the concept of shared reserve power to improve the grid reliability and ensure its dependability.

/pdf/dependable-demand-response-management-in-the-smart-grid-a-vutfkzlv7v.pdf

Dependable Demand Response Management in the Smart Grid: A Stackelberg Game Approach

The Theory of Groups and Quantum Mechanics

A multiresolution approximation (Km)m€Z of L2(R) is of multi- plicity r > 0 if there are r functions x, ... , r whose translates form a Riesz basis for V$ . In the general theory we derive necessary and sufficient conditions for the translates of x, ... , r, y/x, ... , y/r to form a Riesz basis for V\ . The resulting reconstruction and decomposition sequences lead to the construction of dual bases for V0 and its orthogonal complement W0 in Vx . The general theory is applied in the construction of spline wavelets with mul- tiple knots. Algorithms for the construction of these wavelets for some special cases are given.

https://www.ams.org/tran/1994-342-01/S0002-9947-1994-1232187-7/S0002-9947-1994-1232187-7.pdf

WAVELETS OF MULTIPLICITY r

Mallat's construction, via a multiresolution approximation, of orthonormal wavelets generated by a single function is extended to wavelets generated by a finite set of functions. The connection between multiresolution approximation and the concept of wandering subspaces of unitary operators in Hilbert space is exploited in the general setting. An example of multiresolution approximation generated by cardinal Hermite B-splines is constructed

/pdf/wavelets-in-wandering-subspaces-3whqxl9tdu.pdf

Wavelets in wandering subspaces

We show that the Bernstein polynomial basis on a given interval is optimally stable, in the sense that no other nonnegative basis yields systematically smaller condition numbers for the values or roots of arbitrary polynomials on that interval. This result follows from a partial ordering of the set of all nonnegative bases that is induced by nonnegative basis transformations. We further show, by means of some low-degree examples, that the Bernstein form is not uniquely optimal in this respect. However, it is the only optimally stable basis whose elements have no roots on the interior of the chosen interval. These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.

/pdf/on-the-optimal-stability-of-the-bernstein-basis-1vhmh7apx7.pdf

On the optimal stability of the Bernstein basis

Properties of β-splines

We discuss why the variation diminishing property is useful when designing curves or constructing approximation operators from bases with totally positive collocation matrices. Various such bases are considered and a generalisation of the variation diminishing property is presented and applied.

Tim N. T. Goodman

Papers

WAVELETS OF MULTIPLICITY r

Wavelets in wandering subspaces

On the optimal stability of the Bernstein basis

Properties of β-splines

Total Positivity and the Shape of Curves