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 *n*th 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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© Copyright 1961
American Mathematical Society