The Problem of the Completeness of Systems of Particular Solutions of Partial Differential Equations

  • Gaetano Fichera
Part of the ISNM International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique book series (ISNM, volume 49)


Let K be a compact set of the plane of the complex variable z = x + iy and Ω(K) the vector space of complex valued functions f(z) defined on K, continuous on K and holomorphic in any interior point of K (if any). If Ω(K)is endowed with the norm
$$ \left\| {f(z)\left\| {\mathop {\max }\limits_K \left| {f(z)\left| , \right.} \right.} \right.} \right. $$
it becomes a Banach space.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    K. I. WEIERSTRASS, Ueber die analytische Darstellbarkeit sogenannter Funktionen reeller Argumente, Sitzungsb. der K. Preuss Akad. der Wiss. zu Berlin,1885,633–639 and 789–805.Google Scholar
  2. [2]
    C. RUNGE, Zur Theorie der eindeutigen analytischen Funktionen, Acta Math. 6,1885,228–244.Google Scholar
  3. [3]
    E. HILB & O. SZASZ, Allgemeine Reihenentwicklungen, Enzykl. der Math. Wess., 98,124,1129–1276.Google Scholar
  4. [4]
    J. L. WALSH, On the expansion of analytic functions in series of polynomials, Trans. Amer. Math. Soc.,26,1924,155–170.CrossRefGoogle Scholar
  5. [5]
    J. L. WALSH, Ueber die Entwicklung einer analytischen Funktion nach Polynomen, Math. Ann. 96, 1926, 430–436.CrossRefGoogle Scholar
  6. [6]
    J. L. WALSH, Ueber die Entwicklung einer Funktion einer komplexen Varänderliehen nach Polynomen, Math. Ann. 96,1926,437–450.CrossRefGoogle Scholar
  7. [7]
    F. HARTOGS, Ueber die Grenzfunktionen beschränkter Folgen von analytischen Funktionen, Math. Ann. 98,1927,164–178.CrossRefGoogle Scholar
  8. [8]
    F. HARTOGS & A. ROSENTHAL, Ueber Folgen analytischer Funktionen, Math. Ann. 100,1928,212–263. and 104, 1931, 606–610.CrossRefGoogle Scholar
  9. [9]
    M. LAVRENTIEV, On conformai mapping, Proc. Phys. Math. Instit. Steklov (Math. Sect.) 5,1934,159–245 (in Russian).Google Scholar
  10. [10]
    M. LAVRENTIEV, Sur les Fonctions (Tune Variable Complexe Représentable par des Séries de Polynomes, Hermann et Cie, Paris, 1936.Google Scholar
  11. [11]
    O. J. FORREL, On approximation to a mapping function by polynomials, Amer. J. Math. 54, 1932, 571–578.CrossRefGoogle Scholar
  12. [12]
    M. V. KELDYSH, Sur l’approximation des fonctions analytiques dans des domaines fermés, Mat. Sbornik N. S. 8, 50, 1940, 137–148.Google Scholar
  13. [13]
    M. V. KELDYSH, Sur la représentation par des series de polynômes des fonctions d’une variable complexe dans de domaines fermés, Mat. Sbornik N. S. 16,58, 1945, 249–258.Google Scholar
  14. [14]
    S. N. MERGELYAN, On the representation by a series of polynomials on closed sets, Dokl. Akad. Nauk SSSR N. N. 77,1951, 565–568,(in Russian); Amer. Math. Soc. Transi. N. 3,1962, 287–293.Google Scholar
  15. [15]
    S. N. MERGELYAN, Uniform approximation to a function of a complex variable, Uspekhi Mat. Nauk N. S. 7,N. 2,1952,31–122 (in Russian); Amer. Math. Soc. Transi. N. 3,1962,294–391.Google Scholar
  16. [16]
    J. L. WALSH, Interpolation and approximation by rational function in the complex domain, Amer. Math. Soc. Colloquium Publ. XX,New York, 1935.Google Scholar
  17. [17]
    G. FICHERA, Approssimazione uniforme delie funzioni olomorfe mediante funzioni razionali aventi poli semplici prefissati, I,II, Rend. Accad. Naz. Lincei 8,27,1959,193–201 and 317–323.Google Scholar
  18. [18]
    G. FICHERA, Sull’approssimazione uniforme délie funzioni olomorfe con funzioni razionali aventi i poli prefissati, Rend. Accad. Naz. Lincei 8,30,1961,347–350.Google Scholar
  19. [19]
    G. FICHERA, Approximation of analytic functions by rational functions with prescribed poles, Comm. on Pure and Appl. Math. 23, 1970, 359–370.CrossRefGoogle Scholar
  20. [20]
    G. FICHERA, Uniform approximation of continuous functions by rational functions, Ann. di Matem. pura e appl. IV, 84, 1970, 375–386.CrossRefGoogle Scholar
  21. [21]
    S. BERGMAN, The kernel function and conformai mapping, Amer. Math. Soc. Mathem. Surveys V, New York,1950.Google Scholar
  22. [22]
    K. HOFFMAN, Banach spaces of analytic functions, Prentice-Hall, Englewood Cliffs N. J.,1962.Google Scholar
  23. [23]
    P. PORCELLI, Linear spaces of analytic functions, Rand McNally & Co., Chicago, 1966.Google Scholar
  24. [24]
    M. NICOLESCU, Les fonctions polyharmoniques, Hermann et Cie., Paris, 1936.Google Scholar
  25. [25]
    M. PICONE, Sulla convergenza delle successioni di funzioni iperarmoniche, Bull. Mathem. Soc. Roumaine des Sci. 37, 1936, 105–112.Google Scholar
  26. [26]
    P. LAX, A stability theory of abstract differential equations and its applications to the study of local behaviors of solutions of elliptic equations, Comm. on Pure and Appl. Math. 8, 1956, 747–766.CrossRefGoogle Scholar
  27. [27]
    B. MALGRANGE, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier 6, 1956, 271–355.CrossRefGoogle Scholar
  28. [28]
    F. E. BROWDER, Approximation by solutions of partial differential equations, Amer. J. of Mathem. 84,1962, 134–160.CrossRefGoogle Scholar
  29. [29]
    M. PICONE, Appunti di Analisi superiore, Rondinella, Napoli,1940«Google Scholar
  30. [30]
    L. AMERIO, Sull’integrazione dell’equazione in un dominio di connessione qualsiasi, Rend. 1st. Lombardo 78, 1944–45, 1–24.Google Scholar
  31. [31]
    L. AMERIO, Sul calcolo delle soluzioni dei problemi al contorno per le equazioni lineari del secöndo ordine di tipo ellittico, Amer. J. Math. 69,1947,447–489.CrossRefGoogle Scholar
  32. [32]
    G. FICHERA, Teoremi di completezza sulla frontiera di un dominio per taluni sistemi di funzioni, Ann. di Matem. pura e appl. IV,27,1948,1–28.CrossRefGoogle Scholar
  33. [33]
    N. M. GUNTER, Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der Mathematischen Physik, B. G. Teubner,Leipzig, 1957.Google Scholar
  34. [34]
    C. MIRANDA, Sull’approssimazione delle funzioni armoniche in tre variabili, Rend. Acc. Naz. Lincei 8,5, 1948, 530–534.Google Scholar
  35. [35]
    G. FICHERA, Applicazione della teoria del Potenziale di superficie ad aleuni problemi di analisi funzionale lineare,Giorn. di Matern. di Battaglini IV,78,1948–49, 71–80.Google Scholar
  36. [36]
    M. P. COLAUTTI, Sul problema di Neumann per l’equazione ∆2u-λcu=fin un dominio piano a contorno angoloso, Mem. Ace. Sci. Torino 3,4,1959, 1–83.Google Scholar
  37. [37]
    M. P. COLAUTTI, Teoremi di completezza in spazi hilbertiani connessi con 1’equazione di Laplace in duevariabili, Rend. Sem. Matem. di Padova,31,1961, 114–164.Google Scholar
  38. [38]
    G. FICHERA, Numerical and quantitative analysis, Pitman, London, 1978.Google Scholar
  39. [39]
    G. FICHERA, Teoremi di completezza connessi all’integrazione dell’equazione A ∆4, Giorn. di Matem. di Battaglini 77, 1947–48, 184–199.Google Scholar
  40. [40]
    C. MIRANDA, Formole di maggiorazione e teorema di esistenza per le funzioni biarmoniche in due variabili, Giorn. di Mat. di Battaglini 78, 1948–49, 97–118.Google Scholar
  41. [41]
    R. B. ANCORA, Problemi analitici connessi alia teoria della piastra elastica appoggiata, Rend. Sem. Matern. di Padova,20, 1951, 99–134.Google Scholar
  42. [42]
    G. FICHERA, Sui problemi analitici della elasticità piana, Rend. Sem. Matern. Cagliari 18, 1948, 1–22.Google Scholar
  43. [43]
    G. FICHERA, Sull’esistenza e dul calcolo délie soluzioni dei problemi al contorno relativi ail’equilibrio di un corpo elastico, Ann. Scuola Norm. Sup. Pisa, 4, 1950, 35–99.Google Scholar
  44. [44]
    E. MAGENES, Sull’equazione del calore: teoremi di unicità e teoremi di completezza connessi col metodo di integrazione di M. Picone, 1,11, Rend. Sem. Matern. Padova 21, 1952, 94–123 and 136–170.Google Scholar
  45. [45]
    E. MAGENES, Aggiunta aile Note “Sull ‘ equazione del calore; teoremi di unicità, etc. Mimeographed Note, Ist. Mat. Padova,1953.Google Scholar
  46. [46]
    G. FICHERA, Alcuni recenti sviluppi della teoria dei problemi al contorno per le equazioni aile derivate parziali lineari, Convegno Intern. Equaz. Lin. Der. Parz. Trieste 1954, Ed. Cremone+ se 1955, 174–227.Google Scholar
  47. [47]
    C. MIRANDA, Partial differential equations of elliptic type, 2nd. ed. Springer, Berlin-Heidelberg, 1970.CrossRefGoogle Scholar
  48. [48]
    G. FICHERA, On some general integrati on methods employed in connection with linear differential equation, J. of Mathem. and Phys. 29,1950,59–68.Google Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Gaetano Fichera

There are no affiliations available

Personalised recommendations