Nikol’skii inequality between the uniform norm and L q -norm with jacobi weight of algebraic polynomials on an interval

Abstract

We study the Nikol’skii inequality for algebraic polynomials on the interval [−1, 1] between the uniform norm and the norm of the space Lq(α,β), 1 ≤ q −1. We prove that, in the case α > β ≥ −1/2, the polynomial with unit leading coefficient that deviates least from zero in the space Lq(α+1,,β) with the Jacobi weight ϕ(α+1,β)(x) = (1−x)α+1(1+x)β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space Lq(α,β) for 1 ≤ q β ≥ −1/2 is attained.