Characterization of functions in terms of rate of convergence of a quadrature process. II: Case of non-periodic functions (Q1191847)

The authors [Proc. Am. Math. Soc. 111, No. 1, 85-94 (1991; Zbl 0722.41031)] gave characterization of a continuous period function \(f\) in terms of the rate of convergence of the trapezodial rule applied to \(f\) and its translates. It was shown by the authors [Numer. Math. 57, No. 2, 123-138 (1990; Zbl 0693.41031)] that non-periodic functions cannot be characterized in terms of the rate of convergence of the trapezodial rule. In this paper, the authors show that by using a quadrature formula of Lobatto and Radau with respect to the weight \((1-x^ 2)^{-1/2}\) they obtain a quadrature process on \([-1,1]\) which is suitable for characterizing regularity in the non-periodic case.