Advertisement

Abstract

Several authors have used infinite systems of equations for constructions in analysis: Borel, Perron, Riesz, Eidelheit, Pólya, Pittnauer and others, cf. Cooke [2], Pittnauer [13], Schumacher [l5] [l6], Linden-Pittnauer-Wyrwich [8]. Yet there are many more applications available, especially applications concerning multivariate problems. We state some basic principles and results, indicating further lines of research.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Carleman, T.: Sur un théorème de Weierstrass. Ark. Mat. Astr. Fys. 20 B, 1–5 (1927).Google Scholar
  2. [2]
    Cooke, R.C.: Infinite matrices and sequence spaces. MacMillan & Co., London: 1950 (pp. 31–35).Google Scholar
  3. [3]
    Efimova, N.S.: Various definitions of double asymptotic expansions, and their equivalence. The problem of obtaining double asymptotic expansions by the substitution method. (Russian.) Applied questions in mathematical analysis (Russian), pp. 24–45. Tul’sk Gos. Ped. Inst., Tula, 1972. Math. Rev. 52#8751.Google Scholar
  4. [4]
    Efimova, N.S.: The problem of obtaining double asymptotic expansions by means of operational calculus. (Russian.) Applied questions in mathematical analysis (Russian), pp. 46–60. Tul’sk Gos. Ped. Inst., Tula, 1972. Math. Rev. 52#8752.Google Scholar
  5. [5]
    Eidelheit, M.: Zur Theorie der Systeme linearer Gleichungen. I, II Studia Math. 6, 139–148 (1936);Google Scholar
  6. [5a]
    Eidelheit, M.: Zur Theorie der Systeme linearer Gleichungen. I, II Studia Math. 7, 150–154 (1937).Google Scholar
  7. [6]
    Hoischen, L.: A note on the approximation of continuous functions by integral functions. J. London Math. Soc. 42, 351–354 (1967).CrossRefGoogle Scholar
  8. [7]
    Kaplan, W.: Approximation by entire functions. Michigan Math. J. 3, 43–52 (1955).CrossRefGoogle Scholar
  9. [8]
    Linden, H., Pittnauer, F., Wyrwich, H.: Zur Birkhoff-Interpolation ganzer Funktionen. Acta Math. Acad. Scient. Hung. 31, 259–268 (1978).CrossRefGoogle Scholar
  10. [9]
    Lorentz, G.G.: Approximation by incomplete polynomials (Problems and Results). Proceedings Conference Tampa 1976, 298–301.Google Scholar
  11. [10]
    Lorentz, G.G.: Incomplete polynomials of best approximation. Preprint 1977 (14 pp.).Google Scholar
  12. [11]
    Meyer-König, W., Zeller, K.: Matrixtransformationen mit voller Reichweite. L’Enseignement mathématique 15, 233–235 (1969).Google Scholar
  13. [12]
    Niethammer, W., Zeller, K.: Unendliche Gleichungssysteme mit beliebiger rechter Seite. Math. Z. 96, 1–6 (1967).CrossRefGoogle Scholar
  14. [13]
    Pittnauer, F.: Vorlesungen über asymptotische Reihen. Springer, Berlin-Heidelberg-New York: 1972.CrossRefGoogle Scholar
  15. [14]
    Pólya, G.: Eine einfache, mit funktionentheoretischen Aufgaben verknüpfte, hinreichende Bedingung für die Auflösbarkeit eines Systems unendlich vieler linearer Gleichungen. Comm. Math. Helv. 11, 234–252 (1939);CrossRefGoogle Scholar
  16. [14a]
    Pólya, G.: Ergänzung in Trans. Amer. Math. Soc. 50, 129 (1940).CrossRefGoogle Scholar
  17. [15]
    Schumacher, M.: Ein Pólyasches Momentenproblem und umhüllende asymptotische Potenzreihen. Math. Z. 145, 231–234 (1975).CrossRefGoogle Scholar
  18. [16]
    Schumacher, M.: Ober asymptotische Potenzreihen in zwei Veränderlichen und ein zweidimensionales Momentenproblem. Dissertation Dortmund 1976. Math. Rev. 56#16229.Google Scholar
  19. [17]
    Sinclair, A.: A general solution for a class of approximation problems. Pac. J. Math. 8, 857–866 (1958).Google Scholar
  20. [18]
    Treves, F.: Topological vector spaces, distributions and kernels. Academic Press, London-New York: 1967.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • R. Scherer
    • 1
  • K. Zeller
    • 1
  1. 1.Mathematisches InstitutUniversität TübingenTübingenDeutschland

Personalised recommendations