Fourier Approximation and Hausdorff Convergence Original Research Article

For a function f on Rd we consider its Fourier transform Ff and the (integrable) Cesáro averages FMf of suitable truncations of Ff, described by the formula (FMf)(a)=M−d(M−|a1|)+…(M−|ad|)+ (Ff)(a). We study the speed of the convergence F−1FMf→f, (M→∞), under a metric that is somewhere between the L1- and the L∞-metrics. In this metric (which is appropriate to problems of pattern recognition), the distance between two functions is, more or less, the Hausdorff distance between their graphs. We describe a class of functions f for which the distance between F−1FMf and f is O(M−1/2), the fastest rate of converges one can have for discontinuous f.