On the convergence of Filon quadrature
Given \(k\in \mathbb{R}\), it is assumed that the integral \(\int_{-1}^1 e^{\mathbf{i}kg(x)}\pi(x)dx\) can be evaluated for every polynomial \(\pi\). Therefore, a suitable polynomial quadrature rule can be applied to calculate \(Q(f)=\int_{-1}^1 e^{\mathbf{i}kg(x)}f(x)dx\) when \(f\) is holomorphic on \(\mathcal{G}_f\) and \(g\) is holomorphic on \(\mathcal{G}_f\), where \(\mathcal{G}_g\) and \(\mathcal{G}_f\) are neighborhoods of \([-1,1]\), with \(\mathcal{G}_g \subset \mathcal{G}_f\). Let \(\triangle=\{-1,\dots,-1,1,\dots,1\}\) and \(I_\triangle\) be the interpolation operator of Hermite type associated with the endpoints \(\pm 1\), considering that each of the nodes \(\pm 1\) appears \(p\) times. The pure Filon quadrature (PFQ) of order \(n=2p\) is that given by \(Q(I_\triangle(f))\). The paper is concerned with the convergence of this type of quadrature when \(g'(x)\neq 0\) for all \(x\in[-1,1]\), \(g\) and \(1/g'\) are holomorphic on \(\mathcal{G}_g\), and \(1/g'\in L^\infty(\mathcal{G}_g)\). Assuming an additional condition on \(\mathcal{G}_f\) the PFQ is convergent as \(p\to \infty\). The article shows the advantage of clustering interpolation points near the endpoints \(\pm 1\), so the PFQ represents the extreme case. It is proved that the size of \(\mathcal{G}_f\) plays a role in obtaining convergence of the PFQ as \(n\to \infty\). To create convergent Filon-type quadrature for integrands \(f\) with small domain of analyticity, the author presents two different ways of doing it. One is by considering a composite quadrature, and the other by inserting additional nodes in \([-1,1]\). The numerical stability of the Filon quadrature is not analyzed, however, some examples are given to illustrate the convergence properties.
scientific article; zbMATH DE number 5719641