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_q^(α,β), 1 ≤ q < ∞, with Jacobi weight ϕ^(α,β)(x) = (1 − x)^α(1 + x)^β α ≥ β > −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_q^(α,β), 1 ≤ q < ∞, \(\alpha > \beta \geqslant - \frac{1}{2}\), is attained.