Exact constants in inequalities of the jackson type for quadrature formulas

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.