Asymptotically sharp bounds for the remainder for the best quadrature formulas for several classes of functions

Abstract

For certain classes of functions (all functions are defined on a Jordan measurable set G) defined by a majorant on the modulus of continuity, we find an asymptotically sharp bound for the remainder of an optimal quadrature formula of the form

$$\int_G {f(x)dx \approx \sum\nolimits_{v = 1}^m {c_v f(x^v ).} } $$

When the given majorant of the modulus of continuity is t_α and the nonnegative function P(x) is such that for any nonnegative numbera the set {x∈ G ∶ P(x) ≤ a} is Jordan measurable, then we also find an asymptotically sharp bound for the remainder of an optimal quadrature formula of the form

$$\int_G {P(x)f(x)dx \approx \sum\nolimits_{v = 1}^m {c_v f(x^v ).} } $$