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Exact constants in inequalities of the jackson type for quadrature formulas

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Ukrainian Mathematical Journal Aims and scope

Abstract

We prove that if\(R_n \left( {f,\{ t_k \} ,\{ p_k \} } \right)\) is the error of a simple quadrature formula and ω(ε, δ)1 is the integral modulus of continuity, then, for any δ ≥/π andn,r = 1, 2, …, the following equality is true:\(\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{\omega (f^{(r)} ,\delta )_1 }} = \frac{{\pi \left\| {D_1 } \right\|_\infty }}{{n^r }}\) whereD r is the Bernoulli kernel.

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Authors and Affiliations

  1. Dnepropetrovsk University, Dnepropetrovsk

    V. G. Doronin

  2. Dneprodzerzhinsk Technical University, Dneprodzerzhinsk

    A. A. Ligun

Authors
  1. V. G. Doronin
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  2. A. A. Ligun
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Doronin, V.G., Ligun, A.A. Exact constants in inequalities of the jackson type for quadrature formulas. Ukr Math J 52, 48–54 (2000). https://doi.org/10.1007/BF03029768

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