Optimal quadrature algorithms inH p spaces

Abstract

The problem of finding optimal error quadrature formulas for evaluating $$\int\limits_{ - 1}^1 {f(x)dx} $$ , whereHp and wherep>1, has been investigated in some recent papers (e.g. [1, 3, 4, 10]). In this paper we study a class of `almost optimal' quadrature formulas which were introduced by Stenger [8---11], for computing the integral $$\int\limits_{w^{ - 1} ([ - 1,1])} {f(z)dz,f \in H_p (D).} $$ . HereD is a simply connected domain in the complex plane ? andw is a conformal map ofD onto the unit discU. The cost of our quadratures to obtain an ?-approximation to the above integral is at most twice as much as the cost using the optimal formulas.