Polynomials with laguerre weights in L^p

Abstract

For each p (0 < p ≤ ∞), s ≥ 0, and integer m ≥ 1 we consider the extremal problem

$$E_{s,m,p} : = \inf \left\{ {\left\| {t^s e^{ - t} \left( {t^m - q_{m - 1} \left( t \right)} \right)} \right\|_L p:q_{m - 1} \in p_{m - 1} } \right\},$$

where the L^p-norm is taken over [0, ∞) and p_m−1 is the collection of polynomials of degree at most m−1. The asymptotic behavior of E_s,m,p as n:=s+m → ∞ and s/n → ϑ is determined along with the zero distribution for the associated Chebyshev polynomials. The paper includes the proofs of results announced in [7].