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

<p>The Markov brothers’ inequalities in approximation theory concern polynomials p of degree n and assert bounds for the kth derivatives <font size="2"><span style="font-size:10pt;">|p^{(k)}|</span></font>, 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 <font size="2"><span style="font-size:10pt;">|p^{(k)}|</span></font>. For the problem to be meaningful, additional restrictions on p must be imposed, for example, <font size="2"><span style="font-size:10pt;">p(-1)=p'(-1)= . . . = p^{(k-1)}(-1)=0</span></font>. 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 <font size="2"><span style="font-size:10pt;">p^{(k)}</span></font> on a given set of <font size="2"><span style="font-size:10pt;">n-k+1 </span></font>distinct points in [−1, 1]. Analytic solutions are presented and their fine structure analyzed by computation.</p>