On weak tractability of the Smolyak algorithm for approximation problems

JOURNAL OF APPROXIMATION THEORY

卷:192

Article

Xu, Guiqiao

关键词: Weak tractability;    Smolyak algorithm;    Infinitely differentiable function class;    Standard information;   

DOI  :  10.1016/j.jat.2014.10.016

来源: Elsevier

【 摘 要 】

We consider the problems of L-P-approximation of d-variate analytic functions defined on the cube with directional derivatives of all orders bounded by 1. For 1 <= p < infinity, it is shown that the Smolyak algorithm based on polynomial interpolation at the extrema of the Chebyshev polynomials leads to weak tractability of these problems. This gives an affirmative answer to one of the open problems raised recently by Hinrichs et al. (2014). Our proof uses the polynomial exactness of the algorithm and an explicit bound on the operator norm of the algorithm. (C) 2014 Elsevier Inc. All rights reserved.

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