Derivatives of bivariate polynomials, Markov's theorem and Geronimus nodes
Speaker: L. Harris (Kentucky)
Time: 3:00PM
Date: Tue 14th February 2012
Location: Mathematical Sciences Seminar Room (Ag 1.01)
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 Pm(R2) denote the space of all polynomials p(s,t) of degree at most m and let Nk={(cos(nπ/m),cos(qπ/m)): n−q=kmod2, 0≤n,q≤m}. Then to prove the Markov theorem it suffices to show that the maximum of the values |Dˆkp(1,1)(1,−1)| over polynomials p in Pm(R2) satisfying |p(x)|≤1 for all x in the set Nk of nodes is attained when p(s,t)=Tm(s), where Tm 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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