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Approximation Spaces in the Numerical Analysis of Operator Equations

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Abstract

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.

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References

  1. J.M. Almira and U. Luther, Approximation algebras and applications, in: Trends in Approximation Theory, eds. K. Kopotun, T. Lyche and M. Neamtu (Vanderbilt Univ. Press, Nashville, 2001).

    Google Scholar 

  2. J.M. Almira and U. Luther, Generalized approximation spaces and applications, Math. Nachr., accepted for publication.

  3. J.M. Almira and U. Luther, Compactness and generalized approximation spaces, Numer. Funct. Anal. Optim. 23 (2002) 1–38.

    Google Scholar 

  4. D. Berthold, W. Hoppe and B. Silbermann, A fast algorithm for solving the generalized airfoil equation, J. Comput. Appl. Math. 43 (1992) 185–219.

    Google Scholar 

  5. M.R. Capobianco, The stability and the convergence of a collocation method for a class of Cauchy singular integral equations, Math. Nachr. 162 (1993) 45–58.

    Google Scholar 

  6. M.R. Capobianco, G. Criscuolo and P. Junghanns, A fast algorithm for Prandtl's integro-differential equation, J. Comput. Appl. Math. 77 (1997) 103–128.

    Google Scholar 

  7. M.R. Capobianco, G. Criscuolo, P. Junghanns and U. Luther, Uniform convergence of the collocation method for Prandtl's integro-differential equation, ANZIAM J. 42 (2000) 151–168.

    Google Scholar 

  8. M.R. Capobianco, G. Criscuolo and A. Volpe, Numerical analysis of the collocation method for some integral equations with logarithmic perturbation kernel, Rendiconti del Circolo Matematico di Palermo, Serie II Suppl. 68 (2002) 339–350.

    Google Scholar 

  9. M.R. Capobianco, P. Junghanns, U. Luther and G. Mastroianni, Weighted uniform convergence of the quadrature method for Cauchy singular integral equations, in: Singular Integral Operators and Related Topics, eds. A. Böttcher and I. Gohberg, Operator Theory Advances and Applications, Vol. 90 (Birkhäuser, Basel, 1996) pp. 153–181.

    Google Scholar 

  10. M.R. Capobianco and M.G. Russo, Uniform convergence estimates for a collocation method for the Cauchy singular integral equation, J. Integral Equations Appl. 9 (1997) 1–25.

    Google Scholar 

  11. M. Chen, Z. Chen and G. Chen, Approximate Solutions of Operator Equations (World Scientific, Singapore, 1997).

    Google Scholar 

  12. Z. Chen and Y. Xu, The Petrov-Galerkin and iterated Petrov-Galerkin methods for second-kind integral equations, SIAM J. Numer. Anal. 35 (1998) 406–434.

    Google Scholar 

  13. J.A. Cuminato, Uniform convergence of a collocation method for the numerical solution of Cauchytype singular integral equations: A generalization, IMA J. Numer. Anal. 12 (1992) 31–45.

    Google Scholar 

  14. J.A. Cuminato, On the uniform convergence of a perturbed collocation method for a class of Cauchy singular integral equations, Appl. Numer. Math. 16 (1995) 417–438.

    Google Scholar 

  15. R.A. DeVore and G.G. Lorentz, Constructive Approximation (Springer, Berlin, 1993).

    Google Scholar 

  16. Z. Ditzian and V. Totik, Moduli of Smoothness (Springer, Berlin, 1987).

    Google Scholar 

  17. Z. Ditzian and V. Totik, Remarks on Besov spaces and best polynomial approximation, Proc. Amer. Math. Soc. 104 (1988) 1059–1066.

    Google Scholar 

  18. D. Elliott, Convergence theorems for singular integral equations, in: Numerical Solution of Integral Equations, ed. M.A. Golberg (Plenum Press, New York, 1990) pp. 309–362.

    Google Scholar 

  19. M.A. Golberg and C.S. Chen, Discrete Projection Methods for Integral Equations (Computational Mechanics Publications, 1997).

  20. P. Junghanns, Product integration for the generalized airfoil equation, in: Beiträge zur angewandten Analysis und Informatik (Shaker, Aachen, 1994) pp. 171–188.

    Google Scholar 

  21. P. Junghanns and U. Luther, Cauchy singular integral equations in spaces of continuous functions and methods for their numerical solution, J. Comput. Appl. Math. 77 (1997) 201–237.

    Google Scholar 

  22. P. Junghanns and U. Luther, Uniform convergence of the quadrature method for Cauchy singular integral equations with weakly singular perturbation kernels, Rendiconti del Circolo Matematico di Palermo, Serie II Suppl. 52 (1998) 551–566.

    Google Scholar 

  23. P. Junghanns and U. Luther, Uniform convergence of a fast algorithm for Cauchy singular integral equations, Linear Algebra Appl. 275/276 (1998) 327–347.

    Google Scholar 

  24. U. Luther, Uniform convergence of polynomial approximation methods for Prandtl's integrodifferential equation, Preprint 99-11, Department of Mathematics, University of Chemnitz, Germany (1999).

  25. U. Luther, Representation, interpolation, and reiteration theorems for generalized approximation spaces, Ann. Mat. Pura Appl., accepted for publication.

  26. U. Luther, Cauchy singular integral operators in weighted spaces of continuous functions, Integral Equations Oper. Theory, submitted.

  27. U. Luther and M.G. Russo, Boundedness of the Hilbert transformation in some weighted Besov spaces, Integral Equations Oper. Theory 36 (2000) 220–240.

    Google Scholar 

  28. G. Mastroianni, M.G. Russo and W. Themistoclakis, The boundedness of the Cauchy singular integral operator in weighted Besov-type spaces with uniform norms, Integral Equations Oper. Theory 42 (2002) 57–89.

    Google Scholar 

  29. G. Monegato and S. Prössdorf, Uniform convergence estimates for a collocation and discrete collocation method for the generalized airfoil equation, in: Contribution to Numerical Mathematics, ed. A.G. Agarval (World Scientific, Singapore, 1993) pp. 285–299; see also the errata corrige in the Internal Report, No. 14, Dipartimento di Matematica, Politecnico di Torino (1993).

    Google Scholar 

  30. G. Monegato and L. Scuderi, Weighted Sobolev-type spaces and numerical methods for 1D integral equations, Rendiconti del Circolo Matematico di Palermo, Serie II Suppl. 68 (2002) 95–109.

    Google Scholar 

  31. A. Pietsch, Approximation spaces, J. Approx. Theory 32 (1981) 115–134.

    Google Scholar 

  32. S. Prössdorf and B. Silbermann, Numerical Analysis for Integral and Related Operator Equations (Akademie Verlag, Berlin, 1991).

    Google Scholar 

  33. G. Vainikko, Multidimensional Weakly Singular Integral Equations, Lecture Notes in Mathematics, Vol. 1549 (Springer, New York, 1993).

    Google Scholar 

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  1. Technische Universität Chemnitz, Fakultät für Mathematik, D-09107, Chemnitz, Germany

    U. Luther

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Luther, U. Approximation Spaces in the Numerical Analysis of Operator Equations. Advances in Computational Mathematics 20, 129–147 (2004). https://doi.org/10.1023/A:1025806612592

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