On optimal inequalities between three-point quadratures

Abstract

We examine the family of all (at most) three-point symmetric quadratures on $$[-1,1]$$ [ - 1 , 1 ] which are exact on polynomials of order 3 to find all possible inequalities between them in the class of 3-convex functions. Next we optimise them by using convex combinations of the quadratures considered. We find the optimal quadrature and use it to construct the adaptive method of approximate integration. An effective method to estimate the error of this method is also given. It needs a considerably fewer number of subdivisions of the interval of integration than the classical adaptive methods as well as the method developed by the second-named author in his recent paper (Wa̧sowicz in Aequ Math 94(5):887–898, 2020).

Adaptive Integration of Convex Functions of One Real Variable

“…In this present paper we return to the idea of the so-called stopping inequalities. We developed it for 3-convex functions individually in [5] or in the cooperation with Komisarski in [3]. It should be noticed that the precursors in the field were Clenshaw and Curtis [1] and Rowland and Varol [4].…”