Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets

In a previous paper, we proposed a general Fourier method which 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 the asymptotic expansion of the approximation error for biorthonormal wavelets, as the scale tends to zero. One of the contributions of this paper is the computation of sharp, asymptotically optimal upper bounds for the least-squares approximation error. Another contribution is the application of these results to B-splines and Daubechies scaling functions, which yields explicit asymptotic developments and upper bounds. Thanks to these explicit expressions, we can quantify the improvement that can be obtained by using B-splines instead of Daubechies wavelets. In other words, we can use a coarser spline sampling and achieve the same reconstruction accuracy as Daubechies: Specifically, we show that this sampling gain converges to pi as the order tends to infinity.