Exact constants in inequalities of the jackson type for quadrature formulas

Authors

  • V. G. Doronin
  • A. A. Ligun

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.

Published

25.01.2000

Issue

Section

Research articles

How to Cite

Doronin, V. G., and A. A. Ligun. “Exact Constants in Inequalities of the Jackson Type for Quadrature Formulas”. Ukrains’kyi Matematychnyi Zhurnal, vol. 52, no. 1, Jan. 2000, pp. 46-51, https://umj.imath.kiev.ua/index.php/umj/article/view/4395.