Galerkin Method with Graded Meshes for Wiener-Hopf Operators with PC Symbols in Lp Spaces

  • Pedro A. SantosEmail author
Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 221)


This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in.T here is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.


Galerkin method graded meshes Wiener-Hopf operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M.A. Bastos, P.A. Lopes, and A. Moura Santos. The two straight line approach for periodic diffraction boundary-value problems. J. Math. Anal. Appl., 338 (1):330-349, 2008.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    A. Böttcher, Yu.I. Karlovich, and I. Spitkovsky. Convolution operators and factorization of almost periodic matrix functions, volume 131 of Oper. Theory Adv. Appl. Birkhoäuser, Basel, 2002.CrossRefGoogle Scholar
  3. 3.
    A. Böttcher and B. Silbermann. Analysis of Toeplitz operators. Springer-Verlag, Berlin, second edition, 2006.zbMATHGoogle Scholar
  4. 4.
    L. Castro, F.-O. Speck, and F.S. Teixeira. On a class of wedge diffraction problems posted by Erhard Meister. In Operator theoretical methods and applications to mathematical physics, volume 147 of Oper. Theory Adv. Appl., pages 213-240. Birkhäuser, Basel, 2004.CrossRefGoogle Scholar
  5. 5.
    J. Elschner. On spline approximation for a class of non-compact integral equations. Math. Nachr., 146:271-321, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    R. Hagen, S. Roch, and B. Silbermann. Spectral theory of approximation methods for convolution equations. Birkhäuser, Basel, 1995.CrossRefGoogle Scholar
  7. 7.
    A. Karlovich, H. Mascarenhas, and P.A. Santos. Finite section method for a Banach algebra of convolution type operators on Lp(R) with symbols generated by PC and SO. Integral Equations Operator Theory, 67 (4):559-600, 2010.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    A.V. Kozak. A local principle in the theory of projection methods. Dokl. Akad. Nauk SSSR, 212:1287-1289, 1973. (in Russian; English translation in Soviet Math. Dokl.14:1580-1583, 1974).MathSciNetGoogle Scholar
  9. 9.
    E. Meister, P.A. Santos, and F.S. Teixeira. A Sommerfeld-type diffraction problem with second-order boundary conditions. Z. Angew. Math. Mech., 72 (12):621-630, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    S. Pröassdorf and B. Silbermann. Numerical analysis for integral and related operator equations. Birkhäuser Verlag, Basel, 1991.Google Scholar
  11. 11.
    S. Roch, P.A. Santos, and B. Silbermann. Non-commutative Gelfand theories. Springer, 2010.Google Scholar
  12. 12.
    P.A. Santos and B. Silbermann. Integral Equations Operator Theory, 38 (1):66-80, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    P.A. Santos and B. Silbermann. An approximation theory for operators generated by shifts. Numer. Funct. Anal. Optim., 27 (3-4):451-484, 2006.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    P.A. Santos and F.S. Teixeira. Sommerfeld half-plane problems with higher-order boundary conditions. Math. Nachr., 171:269-282, 1995.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Departamento de Matemática Instituto Superior TécnicoUniversidade Técnica de LisboaLisboaPortugal

Personalised recommendations