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.
Similar content being viewed by others
References
V. P. Motomyi, “On the best quadrature formula for periodic differentiable functions,” in:Approximation of Functions and Summationof Series [in Russian], Dnepropetrovsk University, Dnepropetrovsk (1991), pp. 37–40.
V. G. Doronin and A. A. Ligun, “On the exact constants in Jackson’s type inequalities in the spaceL 2,”E. J. Approxim.,1, No. 2, 189–196 (1995).
V. G. Doronin and A. A. Ligun, “Exact solution of certain extremal problems on classes of functions defined by integral moduli ofcontinuity,”Dokl. Akad. Nauk SSSR, Ser. Mat.,251, No. 1, 1233–1236 (1980).
N. P. Korneichuk, V. F. Babenko, and A. A. Ligun,Extremal Properties of Polynomials and Splines [in Russian], Naukova Dumka, Kiev (1992).
N. P. Korneichuk,Extremal Problems in Approximation Theory [in Russian] Nauka, Moscow 1976.
V. P. Motornyi, “On best quadrature formulas of the type\(\sum {_{\lambda = 1}^n } p_k f(x_\lambda )\) for some classes of periodic differentiable functions,”Izv. Akad. Nauk SSSR, Ser. Mat.,38, No. 3, 583–614 (1974).
A. A. Zhensykbaev, “Best quadrature formulas for certain classes of periodic differentiable functions,”Izv. Akad. Nauk SSSR, Ser.Mar.,41. No. 5, 1110–1124 (1977).
B. D. Boyanov, “Description and existence of optimal quadrature formulas for certain classes of differentiable functions,”Dokl. Akad. Nauk SSSR, Ser. Mat.,232, No. 6, 1233–1236 (1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF03029768