Multivariate Band-Limited Functions: Sampling Representation and Approximation

Abstract

Let G be an arbitrary subset of Rⁿ. A tempered distribution f ∈S′ (Rⁿ) is said to be band-limited to G if the support of the Fourier transform of f is contained in G. In several aspects of the harmonic approximation for functions defined on Rⁿ, functions which are band-limited to G play a basic role as multivariate trigonometric polynomials for multivariate periodic functions (cf. [3,4,5,6,7]). Here G may be of various shape and usually has finite measure. Let us consider a simple example. Let W be the set of all those functions defined on R² such that the mixed derivatives ∂³f/∂xy² and ∂ ³f/∂x²y are L₂-bounded with 1. Then the so-called smooth hyperbolic cross G(t) = {(x,y) ∈R²: max (|xy²|,|x²y|) ≤ t }, meas G(t) = N, is optimal for the best L₂-approximation of W by sets B_G,2 of functions f ∈L₂(R²) which are band-limited to G, where mess G is not greater than N. This is one side of background of our paper.