A Smirnov-Bickel-Rosenblatt theorem for compactly-supported wavelets

In nonparametric statistical problems, we wish to find an estimator of an unknown function f. We can split its error into bias and variance terms; Smirnov, Bickel and Rosenblatt have shown that, for a histogram or kernel estimate, the supremum norm of the variance term is asymptotically distributed as a Gumbel random variable. In the following, we prove a version of this result for estimators using compactly-supported wavelets, a popular tool in nonparametric statistics. Our result relies on an assumption on the nature of the wavelet, which must be verified by provably-good numerical approximations. We verify our assumption for Daubechies wavelets and symlets, with N = 6, ..., 20 vanishing moments; larger values of N, and other wavelet bases, are easily checked, and we conjecture that our assumption holds also in those cases

Honest adaptive confidence bands and self-similar functions

Confidence bands are confidence sets for an unknown function f, containing all functions within some sup-norm distance of an estimator. In the density estimation, regression, and white noise models, we consider the problem of constructing adaptive confidence bands, whose width contracts at an optimal rate over a range of H\"older classes. While adaptive estimators exist, in general adaptive confidence bands do not, and to proceed we must place further conditions on f. We discuss previous approaches to this issue, and show it is necessary to restrict f to fundamentally smaller classes of functions. We then consider the self-similar functions, whose H\"older norm is similar at large and small scales. We show that such functions may be considered typical functions of a given H\"older class, and that the assumption of self-similarity is both necessary and sufficient for the construction of adaptive bands. Finally, we show that this assumption allows us to resolve the problem of undersmoothing, creating bands which are honest simultaneously for functions of any H\"older norm

Buckling of bar by wavelet galerkin method

Wavelet are used in data compression, signal analysis and image processing and also for analyzing non stationary time series. Wavelets are the functions which satisfy certain mathematics requirements and are used in other functions. The use of wavelets in mechanics can be viewed from two perspectives, first the analysis of mechanical response for extraction of modal parameters, damage measures, de-noising etc and second the solution of the differential equations governing the mechanical system. Wavelet theory provides various basis functions and multi-resolution methods for finite element method. Wavelet-based beam element can be constructed by using Daubechies scaling functions as an interpolating function. In the present thesis, the compactly supported Daubechies wavelet based numerical solution of boundary value problem has been presented for the instability analysis of prismatic members. This problem can be discretized by the Wavelet-Galerkin method. The evaluation of connection coefficients plays an important role in applying wavelet galerkin method to solve partial differential equations. The buckling problem of axially compressed bars by using Wavelet-Galerkin method is explained in this thesis. The comparisons are made with analytical solutions and with finite element results. The present investigation indicated that wavelet technique provides a powerful alternative to the finite element method. The thesis has been presented in six number of chapters. Chapter 1 deals with the general introduction to wavelets and different types of wavelet families and their properties. The review of literature confining to the scope of study has been presented in chapter 2. The Chapter 3 deals with the properties of Daubechies wavelets and determination of scaling function, wavelet function, filter coefficients and moments of scaling function for different order of Daubechies wavelets. The computation of connection coefficients are described in Chapter 4. Chapter 5 deals with the Wavelet Galerkin method and the buckling problem of prismatic bar by using wavelet Galerkin method. It also includes the numerical results obtained from the present work, and comparisons made with analytical solutions and with finite element results. The Chapter6 concludes the present investigation. An account of possible scope of extended study has been presented to the concluding remarks. At last, some important publications and books referred during the present investigation have been listed in Reference section. KEYWORDS: Introduction to wavelet theory, Daubechies wavelets scaling function, connection coefficients, wavelet galerkin method, buckling of bars

The Multilevel Structures of NURBs and NURBlets on Intervals

This dissertation is concerned with the problem of constructing biorthogonal wavelets based on non-uniform rational cubic B-Splines on intervals. We call non-uniform rational B-Splines ``NURBs , and such biorthogonal wavelets ``NURBlets . Constructing NURBlets is useful in designing and representing an arbitrary shape of an object in the industry, especially when exactness of the shape is critical such as the shape of an aircraft. As we know presently most popular wavelet models in the industry are approximated at boundaries. In this dissertation a new model is presented that is well suited for generating arbitrary shapes in the industry with mathematical exactness throughout intervals; it fulfills interpolation at boundaries as well

Compact support wavelet representations for solution of quantum and electromagnetic equations: Eigenvalues and dynamics

Wavelet-based algorithms are developed for solution of quantum and electromagnetic differential equations. Wavelets offer orthonormal localized bases with built-in multiscale properties for the representation of functions, differential operators, and multiplicative operators. The work described here is part of a series of tools for use in the ultimate goal of general, efficient, accurate and automated wavelet-based algorithms for solution of differential equations. The most recent work, and the focus here, is the elimination of operator matrices in wavelet bases. For molecular quantum eigenvalue and dynamics calculations in multiple dimensions, it is the coupled potential energy matrices that generally dominate storage requirements. A Coefficient Product Approximation (CPA) for the potential operator and wave function wavelet expansions dispenses with the matrix, reducing storage and coding complexity. New developments are required, however. It is determined that the CPA is most accurate for specific choices of wavelet families, and these are given here. They have relatively low approximation order (number of vanishing wavelet function moments), which would ordinarily be thought to compromise both wavelet reconstruction and differentiation accuracy. Higher-order convolutional coefficient filters are determined that overcome both apparent problems. The result is a practical wavelet method where the effect of applying the Hamiltonian matrix to a coefficient vector can be calculated accurately without constructing the matrix. The long-familiar Lanczos propagation algorithm, wherein one constructs and diagonalizes a symmetric tridiagonal matrix, uses both eigenvalues and eigenvectors. We show here that time-reversal-invariance for Hermitian Hamiltonians allows a new algorithm that avoids the usual need to keep a number Lanczos vectors around. The resulting Conjugate Symmetric Lanczos (CSL) method, which will apply for wavelets or other choices of basis or grid discretization, is simultaneously low-operation-count and low-storage. A modified CSL algorithm is used for solution of Maxwell's time-domain equations in Hamiltonian form for non-lossy media. The matrix-free algorithm is expected to complement previous work and to decrease both storage and computational overhead. It is expected- that near-field electromagnetic solutions around nanoparticles will benefit from these wavelet-based tools. Such systems are of importance in plasmon-enhanced spectroscopies

Wavelet Theory

The wavelet is a powerful mathematical tool that plays an important role in science and technology. This book looks at some of the most creative and popular applications of wavelets including biomedical signal processing, image processing, communication signal processing, Internet of Things (IoT), acoustical signal processing, financial market data analysis, energy and power management, and COVID-19 pandemic measurements and calculations. The editor’s personal interest is the application of wavelet transform to identify time domain changes on signals and corresponding frequency components and in improving power amplifier behavior

Frame Theory for Signal Processing in Psychoacoustics

This review chapter aims to strengthen the link between frame theory and signal processing tasks in psychoacoustics. On the one side, the basic concepts of frame theory are presented and some proofs are provided to explain those concepts in some detail. The goal is to reveal to hearing scientists how this mathematical theory could be relevant for their research. In particular, we focus on frame theory in a filter bank approach, which is probably the most relevant view-point for audio signal processing. On the other side, basic psychoacoustic concepts are presented to stimulate mathematicians to apply their knowledge in this field

Multi-Resolution Analysis Using Wavelet Basis Conditioned on Homogenization

This dissertation considers an approximation strategy using a wavelet reconstruction scheme for solving elliptic problems. The foci of the work are on (1) the approximate solution of differential equations using multiresolution analysis based on wavelet transforms and (2) the homogenization process for solving one and two-dimensional problems, to understand the solutions of second order elliptic problems. We employed homogenization to compute the average formula for permeability in a porous medium. The structure of the associated multiresolution analysis allows for the reconstruction of the approximate solution of the primary variable in the elliptic equation. Using a one-dimensional wavelet reconstruction algorithm proposed in this work, we are able to numerically compute the approximations of the pressure variables. This algorithm can directly be applied to elliptic problems with discontinuous coefficients.We also implemented Java codes to solve the two dimensional elliptic problems using our methods of solutions. Furthermore, we propose homogenization wavelet reconstruction algorithm, fast transform and the inverse transform algorithms that use the results from the solutions of the local problems and the partial derivatives of the pressure variables to reconstruct the solutions