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Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in L2(ℝ3)

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Abstract

We study Jackson’s inequality between the best approximation of a function f ∈ L2(ℝ3) by entire functions of exponential spherical type and its generalized modulus of continuity. We prove Jackson’s inequality with the exact constant and the optimal argument in the modulus of continuity. In particular, Jackson’s inequality with the optimal parameters is obtained for classical modulus of continuity of order r and Thue–Morse modulus of continuity of order r ∈ ℕ. These results are based on the solution of the generalized Logan problem for entire functions of exponential type. For it we construct a new quadrature formulas for entire functions of exponential type.

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Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Tula State University, pr. Lenin 92, Tula, 300012, Russia

    Valerii Ivanov & Alexey Ivanov

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  1. Valerii Ivanov
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  2. Alexey Ivanov
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Correspondence to Valerii Ivanov.

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Supported by the Russian Foundation for Basic Research (Grant No. 16-01-00308)

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Ivanov, V., Ivanov, A. Generalized Logan’s Problem for Entire Functions of Exponential Type and Optimal Argument in Jackson’s Inequality in L2(ℝ3). Acta. Math. Sin.-English Ser. 34, 1563–1577 (2018). https://doi.org/10.1007/s10114-018-4437-6

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  • DOI: https://doi.org/10.1007/s10114-018-4437-6

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