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

ISSN 2196-1581

Marco Schuchmann

• Verlag: Books on Demand
• Jahr: 2015
• ISBN drucken:
  9783734744037
• eBook-ISBN:
  9783738669428
• Auflage: 2
• Thema: Mathematik
• Thema: Mathematik Allgemein

Journal of Approximation Theory and Applied Mathematics Vol. 4ContentApproximation Error by Using a Finite Number of BaseCoefficients for Special Types of WaveletsSolving Fredholm Integral Equations with Applicationof the Four Chebyshev PolynomialsFourier Properties of Approximations with Functions on a CompactInterval using Daubechies Wavelets

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 y_j of y is determined by an orthogonal projection from y on V_j 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 ϕ 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 V_j, where V_j belongs to a the sequence of subspaces with the following property:

is an orthonormal basis of V_j with

We use the approximation function

With the function ϕ of the MSA we get an orthonormal basis of V_j with

So we get the orthogonal projection from a L²(R) function f in V_j with

Approximation Error Through a finite k_max

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

Now we consider a father wavelet with compact support: supp

with a > 0.

For Daubechies wavelets of order r we get a = 2r – 1.

Considering Functions with an exponential decay

If

(with d > 0 and c > 0) than we get for k > 0 with

We assume that k_max is so big, that the supp

If we use the maximum |c_k| (with an equality in the equ...

Inhaltsverzeichnis

• Table of 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
• Fourier Properties of Approximations with Functions on a Compact Interval using Daubechies Wavelets
• Copyright