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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