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

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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 L (α,β) q , 1 ≤ q < ∞, with the Jacobi weight ϕ(α,β)(x) = (1 − x)α(1 + x)β, α ≥ β > −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 L (α+1,,β) q 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 L (α,β) q for 1 ≤ q < ∞ and α > β ≥ −1/2 is attained.

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Authors and Affiliations

  1. Institute of Mathematics and Computer Science, Ural Federal University, pr. Lenina 51, Yekaterinburg, 620000, Russia

    V. Arestov & M. Deikalova

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  1. V. Arestov
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  2. M. Deikalova
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Correspondence to M. Deikalova.

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This work was supported by the Russian Foundation for Basic Research (project no. 15-01-02705) and by the Competitiveness Enhancement Program of the Ural Federal University (Enactment of the Government of the Russian Federation no. 211, agreement no. 02.A03.21.0006).

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Arestov, V., Deikalova, M. Nikol’skii inequality between the uniform norm and L q -norm with jacobi weight of algebraic polynomials on an interval. Anal Math 42, 91–120 (2016). https://doi.org/10.1007/s10476-016-0201-2

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