Quantitative L2 error analysis for interpolation methods and wavelet expansions

Our goal in this paper is to set a theoretical basis for the comparison of re-sampling and interpolation methods. We consider the general problem of the approximation of an arbitrary continuously-defined function f(x)-not necessarily bandlimited-when we vary the sampling step T. We present an accurate L² computation of the induced approximation error as a function of T for a general class of linear approximation operators including interpolation and other kinds of projectors. This new quantitative result provides exact expressions for the asymptotic development of the error as T→0, and also sharp (asymptotically exact) upper bounds