On the L₂ Markov Inequality with Laguerre Weight

الأصل · EN

Let wα(t)=tαe⁻ᵗ, α>-1, be the Laguerre weight function, and |·|wα denote the associated L₂-norm, i.e., | f|wα:=(∫₀∞wα(t)| f(t)|²dt)¹/². Denote by 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∈ Pₙ, namely the constant cₙ(α)=p∈ Pₙₚ≠ ₀| p′|wα| p|wα, and we are also interested in its asymptotic value c(α)=ₙ→∞cₙ(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.