2016/05/09 by Nikolov, Geno, Shadrin, Alexei
#41A17 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1605.02508
Let wα(t)=tα e-t, α>-1, be the Laguerre weight function, and |⋅|wα denote the associated L2-norm, i.e., | f|wα:=(∫0∞wα(t)| f(t)|2 dt)1/2. Denote by \cal Pn the set of algebraic polynomials of degree not exceeding n. We study the best constant cn(α) in the Markov inequality in this norm, | p′|wα≤ cn(α) | p|wα , p∈ \cal Pn , namely the constant cn(α)=sup_\mathop^p∈ \cal Pnp≠ 0\frac| p′|wα| p|wα , and we are also interested in its asymptotic value c(α)=limn→∞\fraccn(α)n . In this paper we obtain lower and upper bounds for both cn(α) and c(α). % Note that according to a result of P. Dörfler from 2002, c(α)=[j(α-1)/2,1]-1, with jν,1 being the first positive zero of the Bessel function Jν(z), hence our bounds for c(α) imply bounds for j(α-1)/2,1 as well.