On Nikol'skii type inequality between the uniform norm and the integral q-norm with Laguerre weight of algebraic polynomials on the half-line

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 Lαq 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+1q 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 Lαq are studied.