TY - JOUR
T1 - A bivariate Markov inequality for Chebyshev polynomials of the second kind
AU - Harris, Lawrence A.
N1 - Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/12
Y1 - 2011/12
N2 - This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov's theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel-Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.
AB - This note presents a Markov-type inequality for polynomials in two variables where the Chebyshev polynomials of the second kind in either one of the variables are extremal. We assume a bound on a polynomial at the set of even or odd Chebyshev nodes with the boundary nodes omitted and obtain bounds on its even or odd order directional derivatives in a critical direction. Previously, the author has given a corresponding inequality for Chebyshev polynomials of the first kind and has obtained the extension of V.A. Markov's theorem to real normed linear spaces as an easy corollary.To prove our inequality we construct Lagrange polynomials for the new class of nodes we consider and give a corresponding Christoffel-Darboux formula. It is enough to determine the sign of the directional derivatives of the Lagrange polynomials.
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U2 - 10.1016/j.jat.2011.07.001
DO - 10.1016/j.jat.2011.07.001
M3 - Article
AN - SCOPUS:80054701890
SN - 0021-9045
VL - 163
SP - 1806
EP - 1814
JO - Journal of Approximation Theory
JF - Journal of Approximation Theory
IS - 12
ER -