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Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials

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Abstract

Let υ be a weight on (−1, 1), i.e., a measurable integrable nonnegative function nonzero almost everywhere on (−1, 1). Denote by Lυ(−1, 1) the space of real-valued functions f integrable with weight υ on (−1, 1) with the norm \(f = \int\limits_{ - 1}^1 {f(x)v(x)dx.} \). We consider the problems of the best one-sided approximation (from below and from above) in the space Lυ(−1, 1) to the characteristic function of an interval (a, b), −1 < a < b < 1, by the set of algebraic polynomials of degree not exceeding a given number. We solve the problems in the case where a and b are nodes of a positive quadrature formula under some conditions on the degree of its precision as well as in the case of a symmetric interval (−h, h), 0 < h < 1, for an even weight υ.

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Acknowledgments

The authors are grateful to Professor V.V. Arestov for his attention to their research and useful discussion of the results.

Funding

This work was supported by the Russian Foundation for Basic Research (project no. 18- 01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

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Authors and Affiliations

  1. Ural Federal University, Yekaterinburg, 620000, Russia

    M. V. Deikalova & A. Yu. Torgashova

Authors
  1. M. V. Deikalova
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  2. A. Yu. Torgashova
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Correspondence to M. V. Deikalova or A. Yu. Torgashova.

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Russian Text © The Author(s), 2018, published in Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Vol. 24, No. 4, pp. 110-125.

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Deikalova, M.V., Torgashova, A.Y. Best One-Sided Approximation in the Mean of the Characteristic Function of an Interval by Algebraic Polynomials. Proc. Steklov Inst. Math. 308 (Suppl 1), 68–82 (2020). https://doi.org/10.1134/S0081543820020066

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