Asymptotically Optimal Sampling Schemes for Periodic Functions II: The Multivariate Case

Abstract

We continue with the theme studied in [2] and examine certain questions pertaining to best methods to recover classes of periodic functions from function evaluations. The main feature that distinguishes this work from [2] is that we are concerned with functions of more than one variable, and this leads us to difficulties not encountered in [2]. We treat periodic function classes similar to those studied in [2], which, because of their greater generality, include those bivariate spaces appearing in [7]. Nevertheless, we follow [7] and [3,4,1] and show how blending interpolation operators yield better error estimates, per function evaluation, than the available tensor product methods. Surprisingly, however, we will also demonstrate that tensor products of equally spaced sampling schemes are optimal in a wide variety of cases. Outstanding unresolved problems remain in determining optimal sampling schemes even for standard spaces of periodic functions. This will be mentioned at the end of the paper.