On the Markov inequality in the L₂-norm with Gegenbauer weight

2015/10/12 by Shadrin, Alexei, Nikolov, Geno, Aleksov, Dragomir
#41A17 #41A44 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1510.03265

Abstract

Let w_λ(t)=(1-t²)^λ-1/2, λ>-1/2, be the Gegenbauer weight function, and \Vert⋅\Vert denote the associated L₂-norm, i.e., \Vert f\Vert:=(∫_-1¹w_λ(t)\vert f(t)\vert² dt)^1/2. Denote by P_n the set of algebraic polynomials of degree not exceeding n. We study the best (i.e., the smallest) constant c_n,λ in the Markov inequality \Vert p^′\Vert≤ c_n,λ \Vert p\Vert, p∈ P_n, and prove that c_n,λlt; ((n+1)(n+2λ+1))/(2√(2λ+1)), λgt;-1/2 . Moreover, we prove that the extremal polynomial in this inequality is even or odd depending on whether n is even or odd.

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