On the Rate of Convergence of Difference Schemes for Parabolic Initial-Value Problems and of Singular Integrals

  • Jörgen Löfström
Part of the ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique book series (ISNM, volume 10)


The theory of interpolation spaces has applications to many branches of Analysis, in particular to Approximation theory (see Berens [1] [2], Butzer—Berens [5], Löfström [11] [12], Peetre [15] [18], and others). Our main intention in this paper is to apply the techniques of interpolation spaces (actually disguised as Besov spaces) to some problems related to finite difference approximations for partial differential equations. In doing so we extend and complement previous work by Peetre—Thomée [19], Hedstrom [7], Widlund [24].


Difference Scheme Besov Space Singular Integral Interpolation Space Fourier Multiplier 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. Berens, Approximationssätze für Halbgruppenoperatoren in intermediären Räumen. Schriftenreihe Math. Inst. Univ. Münster 32 (1964).Google Scholar
  2. [2]
    H. Berens, Interpolationsmethoden zur Behandlung von Approximationsprozessen auf Banachräumen. Lecture Notes in Mathematics 64, Springer, Berlin 1968.CrossRefGoogle Scholar
  3. [3]
    O. V. Besov, Investigations of a family of function spaces in connection with theorems of embedding and extension. Trudy. Mat. Inst. Steklov. 60 (1961), 42–81.Google Scholar
  4. [4]
    P. L. Butzer, Fourier-transform methods in the theory of approximation. Arch. Rational Mech. Anal. 5 (1960), 390–415.CrossRefGoogle Scholar
  5. [5]
    P. L. Butzer and H. Berens, Semi-groups of Operators and Approximation. Springer, Berlin 1967.CrossRefGoogle Scholar
  6. [6]
    J. Favard, Sur l’approximation des fonctions d’une variable réelle. Analyse Harmonique, Colloques Internationaux du Centre National de la Recherche Scientifique. 15 (1949), 97–110.Google Scholar
  7. [7]
    G. W. Hedstrom, The rate of convergence of some difference schemes. Department of Mathematics, Chalmers Institute of Technology and the University of Göteborg, (1967).Google Scholar
  8. [8]
    T. Holmstedt, Interpolation d’espaces quasi-normés. C. R. Acad. Sci. Paris. Ser. A. 264 (1967), 242–244.Google Scholar
  9. [9]
    L. Hörmander, Estimates for translation invariant operators on L p spaces. Acta Math. 104 (1960), 93–140.CrossRefGoogle Scholar
  10. [10]
    J. L. Lions, L. I. Lizorkin and S. M. Nikolskii, Integral representations and isomorphism properties of some classes of functions. Ann. Scoula Norm. Sup. Pisa 19 (1965), 127–178.Google Scholar
  11. [11]
    J. Löfström, Some theorems on interpolation spaces with applications to approximation in L p. Math. Ann. 172 (1967), 176–196.CrossRefGoogle Scholar
  12. [12]
    J. Löfström, Besov spaces in the theory of approximation. Manuscript.Google Scholar
  13. [13a]
    P. L. Butzer and R. J. Nessel, Contributions to the theory of saturation for singular integrals in several variables I. General theory. Nederl. Akad. Wetensch. Indag. Math. 28 (1966), 515–531.Google Scholar
  14. [13b]
    R. J. Nessel, Contributions to the theory of saturation for singular integrals in several variables II. Applications, III. Radial kernels. Nederl. Akad. Wetensch. Indag. Math. 29 (1967), 52–73.Google Scholar
  15. [14]
    J. Peetre, Espaces d’interpolation, généralisations, applications. Rend. Seminar Mat. Fis. Milano 34 (1964), 133–164.CrossRefGoogle Scholar
  16. [15]
    J. Peetre, Applications de la théorie des espaces d’interpolation dans l’analyse harmonique. Ricerche Mat. 15 (1966), 3–36.Google Scholar
  17. [16]
    J. Peetre, Reflections about Besov spaces. Lecture notes Lund (1966). (In Swedish.)Google Scholar
  18. [17]
    J. Peetre, Sur les espaces de Besov. C. R. Acad. Sci. Paris Sér. A. 264 (1967), 281–283.Google Scholar
  19. [18]
    J. Peetre, Operators of finite Riesz order. Department of Mathematics, Lund 1966. (In Swedish.)Google Scholar
  20. [19]
    J. Peetre and V. Thomée, On the rate of convergence for discrete initial-value problems. Department of Mathematics, Chalmers Institute of Technology and the University of Goteborg.(1967).Google Scholar
  21. [20]
    R. D. Richtmyer and K. W. Morton, Difference methods for initial value problems. Interscience Pubi. Inc. New York 1967.Google Scholar
  22. [21]
    H. S. Shapiro, A Tauberian theorem related to approximation theory. Acta Math. 120 (1968), 279–292.CrossRefGoogle Scholar
  23. [22]
    W. G. Strang, Polynomial approximation of Bernstein type. Trans. Amer. Math. Soc. 105 (1962), 525–535.CrossRefGoogle Scholar
  24. [23]
    V. Thomée, Stability of difference schemes in the maximum-norm. J. Differential Equations 1 (1965), 273–292.CrossRefGoogle Scholar
  25. [24]
    O. B. Widlund, On the rate of convergence for parabolic schemes. (In print).Google Scholar
  26. [25]
    O. B. Widlund, Stability of parabolic difference schemes in the maximum norm. Numer. Math. 8 (1966), 186–202.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • Jörgen Löfström
    • 1
  1. 1.Dept. of Math.Lund Institute of TechnologySweden

Personalised recommendations