Scripts for a discrete top-down Markov problem in approximation theory

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By Walter Gautschi

Purdue University

Matlab scripts for a discrete top-down Markov problem in approximation theory

Version 1.0 - published on 04 Oct 2016 doi:10.4231/R74Q7RX0 - cite this Archived on 05 Nov 2016

Licensed under Attribution 3.0 Unported

Description

The Markov brothers’ inequalities in approximation theory concern polynomials p of degree n and assert bounds for the kth derivatives |p^{(k)}|, 1 ≤k ≤n, on [−1, 1], given that |p| ≤1 on [−1, 1]. Here we go the other direction, seeking bounds for |p|, given a bound for |p^{(k)}|. For the problem to be meaningful, additional restrictions on p must be imposed, for example, p(-1)=p'(-1)= . . . = p^{(k-1)}(-1)=0. The problem then has an easy solution in the continuous case, where the polynomial and their derivatives are considered on the whole interval [−1, 1], but is more challenging, and also of more interest, in the discrete case, where one focusses on the values of p and p^{(k)} on a given set of n-k+1 distinct points in [−1, 1]. Analytic solutions are presented and their fine structure analyzed by computation.

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The dataset contains 32 matlab files.

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