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Some new analytical techniques and their application to irregular cases for the third order ordinary linear boundary-value problem


Author: Nathaniel R. Stanley
Journal: Trans. Amer. Math. Soc. 101 (1961), 351-376
MSC: Primary 34.30
DOI: https://doi.org/10.1090/S0002-9947-1961-0130420-4
Erratum: Trans. Amer. Math. Soc. 103 (1962), 559.
Erratum: Trans. Amer. Math. Soc. 102 (1962), 545.
MathSciNet review: 0130420
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Abstract: 1. For the operator $ T_3^ - (D)$ defined by $ - {d^3}/d{x^3}$ and a triple of boundary conditions irregular in the sense of Birkhoff, the reduction of this triple to canonical forms is implicit in the reduction made for a more general third order operator (Theorem 1.2).

2. A new technique is developed for calculating the Green's function for the nth order ordinary linear boundary-value problem (Theorem 2.4), and is applied to $ T_3^ - $; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for $ T_3^ - $ (Theorem 2.6).

3. A new technique is developed for calculating asymptotic expansions for large zeros of exponential sums, and the form of the expansion, which includes a logarithmic asymptotic series, is established by induction (Theorem 3.1); expansions for the cube roots of the eigenvalues of $ T_3^ - $ then follow as special cases.

4. A theorem of Dunford and Schwartz (Theorem 4.0) giving a sufficient condition for completeness of eigenfunctions in terms of growth of the norm of the resolvent operator, is applied to prove that, with a possible exception, the eigenfunctions of $ T_3^ - $ span $ {L_2}(0,1)$ (Theorem 4.5).


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DOI: https://doi.org/10.1090/S0002-9947-1961-0130420-4
Article copyright: © Copyright 1961 American Mathematical Society

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