Quantitative Fourier analysis of approximation techniques: Part II - Wavelets

In a previous paper, we proposed a general Fourier method that provides an accurate prediction of the approximation error, irrespective of the scaling properties of the approximating functions, Here, we apply our results when these functions satisfy the usual two-scale relation encountered in dyadic multiresolution analysis, As a consequence of this additional constraint, the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter, This is, in particular, true for th e asymptotic expansion of the approximation error for biorthonormal wavelet s as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asympto tically optimal upper bounds for the least-squares approximation error. Ano ther contribution is the application of these results to B-splines and Daub echies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify the im provement that can be obtained by using B-splines instead of Daubechies wav elets. In other words, we can use a coarser spline sampling and achieve the same reconstruction accuracy as Daubechies: Specifically, we show that thi s sampling gain converges to pi as the order tends to infinity.