Exact constants in inequalities of the jackson type for quadrature formulas - Doronin, V. G.; Ligun, A. A. (Ukrains’kyi Matematychnyi Zhurnal, 2000)
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: \(\mathop {\inf }\limits_{\{ f_k \} ,\{ p_k \} } \mathop {\sup }\limits_{f \in L_1^r \backslash R_1 } \frac{{\left| {R_n (f,\{ t_k \} ,\{ p_k \} )} \right|}}{{ω (f^{(r)} ,δ )_1 }} = \frac{{π \left\| {D_1 } \right\|_∞ }}{{n^r }}\) whereD r is the Bernoulli kernel.