On the quadrature error in operational quadrature methods for convolutions (Q1203410)

An estimate is derived for the approximation of a Volterra integral operator witator with convolution like kernel by a discrete operator. Let \(T\) be the integral operator defined by \(Tx(t) = \int^ t_{-\infty} k(t-s)x(s)ds\), \(t\in\mathbb{R}\), and let \(T_ h\) be the discrete operator \(T_ hx(nh) = h\sum^ n_{m=0} k(n-m/h)x(mh)\), \(h > 0\). The Laplace transform \(K(s)\) of \(k(t)\) is assumed to be analytic in an obtuse sector \(\Sigma\) of the complex plane and (i) \(K(s) - K(0) = 0(| s|^ \alpha)\), \(| s| \to 0\), \(s\in \Sigma\), (ii) \(K(s) = 0(| s|^ \beta)\), \(| s| \to \infty\), \(s \in \Sigma\), \(\alpha,\beta > 0\). Let \(W^{p,M}(\mathbb{R})\) be the Sobolev space of functions \(x(t)\) with norm \(\| x\|_{W^{p,M}(\mathbb{R})} = \| x\|_{L^ p(\mathbb{R})} + \| x^{(M)}\|_{L^ p(\mathbb{R})}\), \(1 \leq p \leq \infty\). It is proved that \(T\), \(T_ h\) satisfy (iii) \(h^{1/p}\| T_ h x(nh) - Tx(nh)\|_{\ell^ p} \leq C p^ M\| x\|_{L^ p(M)}\) for \(x(t) \in W^{p,M}(\mathbb{R})\). When the weights \(k(n/h)\) are generated from \(K(s)\) by the method of \textit{C. Lubich} [Numer. Math. 52, 413-425 (1988; Zbl 0643.65094)]. An analogous result is proved for \(x(t)\) defined on \(R^ +\).