O I 1 -1 durch die GauB-Quadraturformel Q I n n L w 0 f (x 0) i=1 1 1 Sei Rn : = I - Q das Fehlerfunktional. n Izl1, Fur eine im Kreis Kr I Kr : = {z E a: holomorphe Funktion f, f(z) = L i=O sei f i i = x . ( 1. 1) : = sup{ I a 0 I r i E:JN und R (qo) * O}, qo (x) o 1 n 1 1 In Xr := {f: f holomorph in Kr und Iflr < oo} ist I . I eine Seminorm. Das Fehlerfunktional Rn ist in r (X I· I r) stetig I und fUr II Rn II I r, gilt die Identitat 00 (1 . 2) L i=O Dieser Zugang zu ableitungsfreien Abschatzungen des Fehlerterms (1 3) geht auf Hammerlin [4] zurUck. 15 Erftillt die Gewichtsfunktion w eine der Bedingungen w (t ) w(t ) 1 2 ;;; (1. 4. a) w (-t ) w (-t ) 1 2 beziehungsweise w (t ) w(t ) 1 2 (1. 4. b) ~ w (-t ) w (-t ) 1 2 so gilt mit P (x) (X-X ) . (X-X ) ftir die Fehlernorm 1 n n r 1 Pn(x) (1. 5. a) --,-. . - J w (x) dx Pn(r) -1 r-x beziehungsweise r 1 P (x) (1. 5. b) ( ) J w(x) ~ dx .
Inhaltsverzeichnis
Die Fehlernorm spezieller Gauss-Quadraturformeln. - Solving integral equations on surfaces in space. - An adaptive step size control for Volterra integral equations. - Concerning A(?)-stable mixed Volterra Runge-Kutta methods. - Constrained approximation techniques for solving integral equations. - On the numerical solution by collocation of Volterra integrodifferential equations with nonsmooth solutions. - Inclusion of regular and singular solutions of certain types of integral equations. - Two methods for solving the inverse scattering problem for time-harmonic acoustic waves. - Beyond superconvergence of collocation methods for Volterra integral equations of the first kind. - Optimal discrepancy principles for the Tikhonov regularization of integral equations of the first kind. - Spline-Galerkin method for solving some quantum mechanic integral equations. - Integral treatment of O. D. E with splines. - Product integration for weakly singular integral equations in ? m. - Stability results for discrete Volterra equations: Numerical experiments. - The design of acoustic torpedos. - On the condition number of boundary integral equations in acoustic scattering using combined double- and single-layer potentials. - Numerical solution of singular integral equations and an application to the theory of jet-flapped wings. - Tikhonov-Phillips regularization of the Radon Transform. - Numerical solution of a first kind Fredholm integral equation arising in electron-atom scattering. - Approximate solution of ill-posed equations: Arbitrarily slow convergence vs. superconvergence. - A unified analysis of discretization methods for Volterra-type equations. - Wiener-Hopf integral equations: Finite section approximation and projection methods. . - Stability results for Abel equation. - Problems.
