Numerical Computation of the Markov Factors for the Systems of Polynomials with the Hermite and Laguerre Weights

Abstract

Denote by π _n the set of all real algebraic polynomials of degree at most n, and let $U_{n} := \{e^{-x^2} p(x) : p \in \pi_{n} \}$, $V_{n} := \{e^{-x} p(x) : p \in \pi_{n} \}$. It was $\sup \Vert {\rm I\!R} \in \not \equiv u_{*,n}^{(k)}$ proved in [9] that $M_{k}(U_{n}) := \sup \{ \Vert u^{(k)} \Vert_{{\rm I\!R}}/ \Vert u \Vert_{{\rm I\!R}} : u \in U_{n}, u \not \equiv 0\} = \Vert u_{*,n}^{(k)} \Vert_{{\rm I\!R}},~ \forall n, k \in {\rm I\!N}$, and $M_{k}(V_{n}) := {\rm sup} \{ \Vert v^{(k)} \Vert_{{\rm I\!R}_{+}}/ \Vert v \Vert_{{\rm I\!R}_{+}} : v \in V_{n}, v \not \equiv 0\} = \Vert v_{*,n}^{(k)} \Vert_{{\rm I\!R}_{+}}$, where $\Vert \cdot \Vert_{{\rm I\!R}}$ ($\Vert \cdot \Vert_{{\rm I\!R}_{+}}$) is the supremum norm on IR (IR₊ : = [0, ∞ )) and u _*,n (v _*,n ) is the Chebyshev polynomial from U _n (V _n ). We prove here the convergence of an algorithm for the numerical construction of the oscillating weighted polynomial from U _n (V _n ), which takes preassigned values at its extremal points. As an application, we obtain numerical values for the Markov factors M _k (U _n ) and M _k (V _n ) for 1 ≤ n ≤ 10 and 1 ≤ k ≤ 5.

The research was supported by the Bulgarian Ministry of Education and Science under Contract MM-1402/2004.