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Note on the expansion of the Green's function

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  1. A slight modification of the series must be made in certain cases; see Hilb, loc. cit. p. 87. Ifl is real, the theorems of Hilbert (Gött. Nachr. 1904, p. 49–91 and p. 213–259) and E. Schmidt (Göttingen dissertation 1905; in particular p. 12) ensure the validity of the expansion provided its uniform convergence is taken for granted.

  2. Boundary Value and Expansion Problems of Ordinary Linear Differential Equations, Trans. Am. Math. Soc. 9 (1908), p. 373–395. See also Bull. Am. Math. Soc., 12 (1906–1907), p. 439. Strictly speaking, it is only whenl is unlimitedly differentiable that the above expansion comes directly as a special case of my theorems; nevertheless this stronger assumption onl was not used by me in this connection, but only in discussing certain asymptotic series to terms ofall orders.

  3. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen 43 (1911), p. 68–71; see p. 70–71. I may note in passing that in Bounitzky's paper,Sur la fonction de Green des équations différentielles, linéaires ordinaires, J. de Math. (6) 5 (1909), p. 65–125, appear a number of theorems also to be found in my paper. In particular, the notion of adjoint boundary conditions which is central in his paper is carefully developed by me (loc. cit. p. 375–377).

  4. Loc. cit. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen 43 (1911), p. 390.

  5. Loc. cit. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen 43 (1911), p. 382–383.

  6. Loc. cit. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen 43 (1911), p. 383 and p. 388.

  7. Loc. cit. Sitzungsberichte der Physikalisch-Medizinischen Sozietät zu Erlangen 43 (1911), p. 394–395.

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  1. Princeton, (USA)

    George D. Birkhoff

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Birkhoff, G.D. Note on the expansion of the Green's function. Math. Ann. 72, 292–294 (1912). https://doi.org/10.1007/BF01667329

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