Siberian Mathematical Journal

, Volume 44, Issue 6, pp 1027–1038 | Cite as

Convolution Operators on Expanding Polyhedra: Limits of the Norms of Inverse Operators and Pseudospectra

  • E. A. Maximenko


We consider matrix convolution operators with integrable kernels on expanding polyhedra. We study their connection with convolution operators on the cones at the vertices of polyhedra. We prove that the norm of the inverse operator on a polyhedron tends to the maximum of the norms of the inverse operators on the cones, and the pseudospectrum tends to the union of the corresponding pseudospectra. The study bases on the local method adapted to this kind of problems.

convolution operators polyhedra norm of inverse operator pseudospectrum 


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  1. 1.
    Landau H., “The notion of approximate eigenvalues applied to an integral equation of laser theory,” Quart. Appl. Math., 35, 165–171 (April 1977).Google Scholar
  2. 2.
    Reichel L. and Trefethen L. N., “Eigenvalues and pseudo-eigenvalues of Toeplitz matrices,” Linear Algebra Appl., 162, 153–185 (1992).Google Scholar
  3. 3.
    Böttcher A., “Pseudospectra and singular values of large convolution operators,” J. Integral Equations Appl., 6, 267–301 (1994).Google Scholar
  4. 4.
    Böttcher A. and Wolf H., “Spectral approximation for Segal-Bargmann space Toeplitz operators,” Linear Operators, Banach Center Publ (Warsaw), 387, 25–48 (1997).Google Scholar
  5. 5.
    Grudsky S. M. and Kozak A. V., “On the convergence rate for norms of operators inverse to truncated Toeplitz operators,” in: Integrodi_erential Equations and Some of Their Applications [in Russian], Rostov Univ., Rostov-on-Don, 1995, pp. 45–55.Google Scholar
  6. 6.
    Böttcher A., Grudsky S. M., and Silbermann B., “Norms of inverses, spectra, and pseudospectra of large truncated Wiener-Hopf operators and Toeplitz matrices,” New York J. Math., 3, 1–31 (1997).Google Scholar
  7. 7.
    Böttcher A. and Silbermann B., Introduction to Large Truncated Toeplitz Matrices, Springer-Verlag, New York (1999)Google Scholar
  8. 8.
    Simonenko I. B., “A new general method for studying linear operator equations analogous to singular integral equations,” Dokl. Akad. Nauk SSSR, 158, 790–793 (1964).Google Scholar
  9. 9.
    Simonenko I. B., “A new general method for studying linear operator equations analogous to singular integral equations. I,” Izv. Akad. Nauk SSSR Ser. Mat., 29, 567–586 (1965).Google Scholar
  10. 10.
    Simonenko I. B., “Convolution operators in cones,” Mat. Sb., 74, 298–313 (1967).Google Scholar
  11. 11.
    Simonenko I. B. and Chin ' Ngok Min ', A Local Method in the Theory of One-Dimensional Singular Integral Equations with Piecewise Continuous Coeffcients. Noethericity [in Russian], Rostov Univ., Rostov-on-Don (1986).Google Scholar
  12. 12.
    Simonenko I. B., “The theory of local-type operators and some of their applications,” submitted to VINITI on January 23, 1996, No. 275–B96.Google Scholar
  13. 13.
    Kozak A. V., “The local principle in the theory of projection methods,” Dokl. Akad. Nauk SSSR, 212, 1287–1289 (1973).Google Scholar
  14. 14.
    Kozak A. V., “On the reduction method for discrete multidimensional convolutions,” Mat. Issled. (Kishinëv), 8, 157–160 (1973).Google Scholar
  15. 15.
    Kozak A. V., Projection Methods for Solving Multidimensional Equations of Convolution Type, Diss. Kand. Fiz.-Mat. Nauk, Rostov-on-Don, 1974.Google Scholar
  16. 16.
    Kozak A. V. and Simonenko I. B., “Projection methods for solving multidimensional discrete equations in convolutions,” Sibirsk. Mat. Zh., 21, 119–127 (1980).Google Scholar
  17. 17.
    Krupnik N. Ya., “Exact constant in Simonenko's theorem on the envelope of a family of operators of local type,” Funktsion. Anal. i Prilozhen., 20, 70–71 (1986).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • E. A. Maximenko
    • 1
  1. 1.Rostov State UniversityRostov-on-Don

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