Journal of Approximation Theory and Applied Mathematics - 2014 Vol. 4 E-Book

• Herausgeber: Books on Demand
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Deutsch
• Veröffentlichungsjahr: 2015

Seitenzahl: 21

Beschreibung

Journal of Approximation Theory and Applied Mathematics Vol. 4 Content Approximation Error by Using a Finite Number of Base Coefficients for Special Types of Wavelets Solving Fredholm Integral Equations with Application of the Four Chebyshev Polynomials Fourier Properties of Approximations with Functions on a Compact Interval using Daubechies Wavelets

Leseprobe

Contents

Approximation Error by Using a Finite Number of Base Coefficients for Special Types of Wavelets:

Solving Fredholm Integral Equations with Application of the Four Chebyshev Polynomials:

Approximation Error by Using a Finite Number of Base Coefficients for Special Types of Wavelets

M. Schuchmann and M. Rasguljajew from the Darmstadt University of Applied Sciences

Abstract

If an approximation yj of y is determined by an orthogonal projection from y on Vj then in practical cases only a finite number of bases coefficients can be used. Here we investigate the relationship of the approximation error, resulting from the use of a finite number of basic coefficients, depending on the number of basic elements. We consider wavelets with compact support, such as Daubechies wavelets and the Shannon wavelet for different types of functions.

Introduction of the MSA

In the wavelet theory a scaling function ø is used, which belongs to a MSA (multi scale analysis). From the MSA we know, that we can construct an orthonormal basis of a closed subspace Vj, where Vj belongs to a the sequence of subspaces with the following property:

... ⊂V-1⊂V0⊂V1 ⊂...⊂L2(R),

We use the approximation function

Approximation Error Through a finite kmax

If we use a finite number kmax than we get an approximation error for y in Vj depending on the kmax:

Considering Functions with an exponential decay

We assume that kmax is so big, that the supp øj,-kmax⊆ R-. If we use the maximum |ck| (with an equality in the equation above), we get the following relation between the approximation error and kmax

which is independent from c, m and a. Here we assume additionally that a > kmax and y