Abstract. Let S ⊂ R d be a bounded subset with positive Lebesgue measure. The PaleyWiener space associated to S, P WS, is defined to be the set of all square-integrable functions on R d whose Fourier transforms vanish outside S. A sequence (xj : j ∈ N) in R d is said to be a Riesz-basis sequence for L2(S) (equivalently, a complete interpolating sequence for P WS) if the sequence (e −i x j ,·
Abstract. Given a triangular array of points on [−1, 1] satisfying certain minimal separation conditions, a classical theorem of Szabados asserts the existence of polynomial operators that provide interpolation at these points as well as a near-optimal degree of approximation for arbitrary continuous functions on the interval. This paper provides a simple, functional-analytic proof of this fact. This abstract technique also leads to similar results in general situations where an analogue of the classical Jackson-type theorem holds. In particular, it allows one to obtain simultaneous interpolation and a nearoptimal degree of approximation by neural networks on a cube, radial-basis functions on a torus, and Gaussian networks on Euclidean space. These ideas are illustrated by a discussion of simultaneous approximation and interpolation by polynomials and also by zonal-function networks on the unit sphere in Euclidean space.
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