Jacobi translation and the inequality of different metrics for algebraic polynomials on an interval

Abstract

The sharp inequality of different metrics (Nikol’skii’s inequality) for algebraic polynomials in the interval [−1, 1] between the uniform norm and the norm of the space L (α,β) , 1 ≤ q −1, is investigated. The study uses the generalized translation operator generated by the Jacobi weight. A set of functions is described for which the norm of this operator in the space L (α,β) , 1 ≤ q < ∞, $$\alpha > \beta \geqslant - \frac{1}{2}$$ , is attained.