Error Estimations in an Approximation on a Compact Interval with a Wavelet Bases

Abstract

By an approximation with a wavelet base we have in practice not only an error if the function y is not in V_j. There we have a second error because we do not use all bases functions. If the wavelet has a compact support we have no error by using only a part of all basis function. If we need an approximation on a compact interval I (which we can do even if y is not quadratic integrable on R, because in that case it must only be quadratic integrable on I) leads to worse approximations if we calculate an orthogonal projection from 1_Iy  in V_j. We can get much better approximations, if we apply a least square approximation with points in I. Here we will see, that this approximation can be much better than a orthogonal projection form y or 1_I y in V_j. With the Shannon wavelet, which has no compact support, we saw in many simulations, that a least square approximation can lead to much better results than with well known wavelets with compact support. So in that article we do an error estimation for the Shannon wavelet, if we use not all bases coefficients.