A formula for optimal integration in \(H^ 2\) (Q1802167)

Let \(H^ 2\) be the Hilbert space of the functions which are analytic within the unit disk and square integrable in the Lebesgue sense around the unit circle \(\partial D\), with the scalar product \((f,g) = (1/2\pi)\int_{\partial D}f(z)\overline{g(z)}dz\). For approximating the integral \(If:=\int_ \Gamma f(z)dz\) of a function \(f\in H^ 2\) along a path \(\Gamma \subset \mathbb{C}\), the author considers the integration formula (1) \(Qf = \sum^ n_{j=0}a_ jf_ j\), \(f_ j:= f(z_ j)\), where the \(z_ j\) are points on or outside \(\Gamma\), and \(a_ j \in \mathbb{C}\) are the weights. The author obtains an optimal integration formula of the type (1), i.e. he finds the weights \(a_ j\) for which the error functional \(Ef:= If-Qf\) has minimal norm on \(H^ 2\). He compares numerically the accuracy of the optimal formula with some well-known integration formulae. For points equidistant on a circle of radius \(r\) the formula allows the derivation of the barycentric formula. Numerical examples are considered.