On Nikol'skii Type Inequality between the Uniform Norm and the Integral q-norm with Laguerre Weight of Algebraic Polynomials on the Half-Line

Abstract

We study the Nikol'skii type inequality for algebraic polynomials on the half-line [0,∞) between the “uniform” norm sup{|f(x)|e−x∕2:x∈[0,∞)} and the norm ∫0 ∞|f(x)e−x∕2|qxαdx1∕q of the space Lα q with the Laguerre weight for 1≤q<∞ and α≥0. It is proved that the polynomial with a fixed leading coefficient that deviates least from zero in the space Lα+1 q is the unique extremal polynomial in the Nikol'skii inequality. To prove this result, we use the Laguerre translation. The properties of the norm of the Laguerre translation in the spaces Lα q are studied. © 2017 Elsevier Inc.