L. Harris (Kentucky)

will speak on

Derivatives of bivariate polynomials, Markov's theorem and Geronimus nodes

Time: 3:00PM
Date: Tue 14th February 2012
Location: Mathematical Sciences Seminar Room (Ag 1.01) [map]

Abstract: An outstanding problem that has been recently solved is to prove V. A. Markov's theorem for derivatives of polynomials on any real normed linear space. An elementary argument leads to a reduction of the problem to a certain directional derivative on two dimensional spaces. To state this, let ${\mathcal P}_m(\mathbb R^2)$ denote the space of all polynomials $p(s,t)$ of degree at most $m$ and let
\[
{\mathbb N}_k = \{ (\cos (n\pi/m), \cos (q\pi/m)):~n-q =k\mod 2, ~0\leq n,q
\leq m\}.
\]
Then to prove the Markov theorem it suffices to show that the maximum of the values $|\hat{D}^k p(1,1)(1,-1)|$ over polynomials $p$ in ${\mathcal P}_m(\mathbb R^2)$ satisfying $|p(x)| \leq 1$ for all $x$ in the set ${\mathbb N}_k$ of nodes is attained when $p(s,t) = T_m(s)$, where $T_m$ is the Chebyshev polynomial of degree $m$. We consider more general sets of nodes, called Geronimus nodes, where the extremal polynomials sought are orthogonal polynomials satisfying a three-term recurrence relation with constant coefficients. For example, this includes the Chebyshev polynomials of kinds 1-4.

In the course of our discussion we obtain an explicit formula for Lagrange polynomials and a Lagrange interpolation theorem for the Geronimus nodes. We also deduce a bivariate cubature formula analogous to Gaussian quadrature.

(This talk is part of the Analysis series.)

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