On the approximation power of convolution-based least squares versus interpolation

There are many signal processing tasks for which convolution-based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine-based methods. The coefficients of the corresponding signal approximations are typically obtained by direct sampling (interpolation or quasi-interpolation) or by using least squares techniques that apply a prefilter prior to sampling. We compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h. Specifically, we review several results in approximation theory with a special emphasis on the Strang-Fix (1971) conditions, which relate the general O(h^L ) behavior of the error to the ability of the representation to reproduce polynomials of degree n=L-1. We use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the L₂-error as h tends to zero. We also propose a new improved L₂-error bound for the least squares case. In the process, we provide all the relevant bound constants for polynomial splines. Some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order. We present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations