Approximation with interpolatory constraints

Abstract

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.

Minimum Sobolev norm interpolation of scattered derivative data

“…If we do not require that the dimension of the polynomial space match exactly the number of 1's in the incidence matrix, then it is possible to guarantee not just existence, but also give explicit algorithms and prove the convergence of the resulting polynomials. In the case of interpolation based on the values of the polynomials alone, this has been observed in a series of papers [5,26,24]. The purpose of this paper is to generalize these results for Birkhoff-like interpolation for the so called diffusion polynomials.…”

Sufficient Conditions for Sampling and Interpolation on the Sphere

“…We want to point out that there are other results about sufficient conditions for L p -MZ and interpolating families of points on compact manifolds, but such conditions do not provide precise constants, see [FM11,FM10,MNSW02,OCP12]. We observe that due to the result mentioned above about minimal L p -MZ (or maximal L p -interpolating) arrays, an array with m L = π L cannot satisfy the conditions of Theorems 1.6,1.5.…”

Covering numbers of isotropic reproducing kernels on compact two‐point homogeneous spaces

“…The next lemma is a direct adaptation of [, Corollary 3.1] which is a consequence of the general Theorem 3.2 of . In particular, [, Theorem 2.1] provides a key property that ensure such adaptation.…”

“…In particular, [, Theorem 2.1] provides a key property that ensure such adaptation. Lemma ([, Corollary 3.1]) Let be a finite set of distinct points. Suppose that , for some and some positive integer n .…”