On the exact evaluation of integrals of wavelets

Wavelet expansions are a powerful tool for constructing adaptive approximations. For this reason, they find applications in a variety of fields, from signal processing to approximation theory. Wavelets are usually derived from refinable functions, which are the solution of a recursive functional equation called the refinement equation. The analytical expression of refinable functions is known in only a few cases, so if we need to evaluate refinable functions we can make use only of the refinement equation. This is also true for the evaluation of their derivatives and integrals. In this paper, we detail a procedure for computing integrals of wavelet products exactly, up to machine precision. The efficient and accurate evaluation of these integrals is particularly required for the computation of the connection coefficients in the wavelet Galerkin method. We show the effectiveness of the procedure by evaluating the integrals of pseudo-splines

Anisotropic Operator Symbols Arising From Multivariate Jump Processes

Abstract.: It is shown that infinitesimal generators $${\mathcal{A}}$$ of certain multivariate pure jump Lévy copula processes give rise to a class of anisotropic symbols that extends the well-known classes of pseudo differential operators of Hörmander-type. In addition, we provide minimal regularity convergence analysis for a sparse tensor product finite element approximation to solutions of the corresponding stationary Kolmogorov equations $${\mathcal{A}}u = f$$ . The computational complexity of the presented approximation scheme is essentially independent of the underlying state space dimensio

Vector Subdivision Schemes for Arbitrary Matrix Masks

Employing a matrix mask, a vector subdivision scheme is a fast iterative averaging algorithm to compute refinable vector functions for wavelet methods in numerical PDEs and to produce smooth curves in CAGD. In sharp contrast to the well-studied scalar subdivision schemes, vector subdivision schemes are much less well understood, e.g., Lagrange and (generalized) Hermite subdivision schemes are the only studied vector subdivision schemes in the literature. Because many wavelets used in numerical PDEs are derived from refinable vector functions whose matrix masks are not from Hermite subdivision schemes, it is necessary to introduce and study vector subdivision schemes for any general matrix masks in order to compute wavelets and refinable vector functions efficiently. For a general matrix mask, we show that there is only one meaningful way of defining a vector subdivision scheme. Motivated by vector cascade algorithms and recent study on Hermite subdivision schemes, we shall define a vector subdivision scheme for any arbitrary matrix mask and then we prove that the convergence of the newly defined vector subdivision scheme is equivalent to the convergence of its associated vector cascade algorithm. We also study convergence rates of vector subdivision schemes. The results of this paper not only bridge the gaps and establish intrinsic links between vector subdivision schemes and vector cascade algorithms but also strengthen and generalize current known results on Lagrange and (generalized) Hermite subdivision schemes. Several examples are provided to illustrate the results in this paper on various types of vector subdivision schemes with convergence rates

Wavelet based QRS detection in ECG using MATLAB

In recent years, ECG signal plays an important role in the primary diagnosis, prognosis and survivalanalysis of heart diseases. Electrocardiography has had a profound influence on the practice of medicine.This paper deals with the detection of QRS complexes of ECG signals using derivativebased/Pan-Tompkins/wavelet transform based algorithms. The electrocardiogram signal contains animportant amount of information that can be exploited in different manners. The ECG signal allows for theanalysis of anatomic and physiologic aspects of the whole cardiac muscle. Different ECG signals fromMIT/BIH Arrhythmia data base are used to verify the various algorithms using MATLAB software.Wavelet based algorithm presented in this paper is compared with the AF2 algorithm/Pan-Tompkinsalgorithms for signal denoising and detection of QRS complexes meanwhile better results are obtained forECG signals by the wavelet based algorithm. In the wavelet based algorithm, the ECG signal has beendenoised by removing the corresponding wavelet coefficients at higher scales. Then QRS complexes aredetected and each complex is used to find the peaks of the individual waves like P and T, and also theirdeviations.Keywords: Electrocardiogram (ECG), AF2 Algorithm, MATLAB, Pan-Tompkins algorithm, WaveletTransform, Denoisin

Transforms, algorithms and applications

Fourier transforms and other related transforms are an essential tool in applications of science, engineering and technology. In fact, much of the work currently being done in mathematics, physics and engineering has its roots in Fourier's pioneering idea of representing an arbitrary function as the sum of a trigonometric series. The main purpose of these notes is to give a brief overview of some Fourier-related transforms, namely: continuous Fourier transform, Fourier series, discrete Fourier transform, fast Fourier transform (FFT),sine and cosine transforms, Z-transform, Laplace transform, windowed Fourier transform, continuous and discrete wavelet transforms. Our aim is simply to present a summary of these transforms and to describe their main properties and possible applications, and so most of the results are presented with no proof.References containing the proofs and other details about the transforms are always indicated