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

摘要

We study the Nikol'skii inequality for algebraic polynomials on the interval [-1, 1] between the uniform norm and the norm of the space L (q) ((alpha,beta)) , 1 a parts per thousand currency sign q < a, with the Jacobi weight I center dot(alpha,beta)(x) = (1 - x) (alpha) (1 + x) (beta) , alpha a parts per thousand yen beta > -1. We prove that, in the case alpha > beta a parts per thousand yen -1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L (q) ((alpha+1,,beta)) with the Jacobi weight I center dot ((alpha+1,beta))(x) = (1-x) (alpha+1)(1+x) (beta) 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 L (q) ((alpha,beta)) for 1 a parts per thousand currency sign q < a and alpha > beta a parts per thousand yen -1/2 is attained.