On the Computation of Wavelet Coeecients Abbreviated Title : Wavelet Coeecients

We consider fast algorithms of wavelet decomposition of a function f when discrete observations of f (suppf [0; 1]) are available. The properties of the algorithms are studied for three types of observation design: the regular design, when the observations f(xi) are taken on the regular grid xi = i=N , i = 1; :::; N ; the case of jittered regular grid, when it is only known that for all 1 i N i=N xi < (i + 1)=N ; the random design case: xi i = 1; :::; N are independent and identically distributed random variables on [0; 1]. We show that these algorithms are in certain sense e cient when the accuracy of approximation is concerned. The proposed algorithms are computationally straightforward: the whole e ort to compute the decomposition is order N for the sample size N . 2