Exact constants in inequalities of the jackson type for quadrature formulas

  • 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
How to Cite
DoroninV. G., and LigunA. A. “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.
Section
Research articles