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 defined by 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 ; a necessary and sufficient condition is given for the identification of degenerate sets of boundary conditions for (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 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 span (Theorem 4.5).

**[1]**G. D. Birkhoff,*On the asymptotic character of the solutions of certain linear differential equations containing a parameter*, Trans. Amer. Math. Soc. vol. 9 (1908) pp. 219-231. MR**1500810****[2]**-,*Boundary-value and expansion problems of ordinary linear differential equations*, Trans. Amer. Math. Soc. vol. 9 (1908) pp. 373-395. MR**1500818****[3]**E. A. Coddington and N. Levinson,*Theory of ordinary differential equations*, New York, McGraw-Hill Book Co., Inc., 1955. MR**0069338 (16:1022b)****[4]**R. Courant and D. Hilbert,*Methods of mathematical physics*, vol. 1, New York, Interscience Publishers, Inc., 1953. MR**0065391 (16:426a)****[5]**N. Dunford and J. T. Schwartz,*Linear operators*. Part II, New York, Interscience Publishers, Inc. (forthcoming). MR**1009163 (90g:47001b)****[6]**A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi,*Higher transcendental functions*, vol. 3, New York, McGraw-Hill Book Co., Inc., 1955, pp. 212-217.**[7]**S. Hoffman,*Second order linear differential operators defined by irregular boundary conditions*, Ph.D. dissertation, Yale, 1957.**[8]**J. W. Hopkins,*Some convergent developments associated with irregular boundary conditions*, Trans. Amer. Math. Soc. vol. 20 (1919), pp. 249-259. MR**1501125****[9]**E. L. Ince,*Ordinary differential equations*, New York, Dover Publications, Inc., 1956. MR**0010757 (6:65f)****[10]**T. Muir,*A treatise on the theory of determinants*, New York, Dover Publications, Inc., 1960, pp. 213-216. MR**0114826 (22:5644)****[11]**F. Riesz and B. Sz-Nagy,*Functional analysis*, New York, Ungar Publishing Co., 1955, pp. 145-151. MR**0071727 (17:175i)****[12]**J. T. Schwartz,*Perturbations of spectral operators and applications*. I, Pacific J. Math. vol. 4 (1954) pp. 415-458. MR**0063568 (16:144b)****[13]**E. Schwengeler,*Geometrisches über die Verteilung der Nullstellen spezieller ganzer Funktionen*(*Exponentialsummen*), Ph.D. dissertation, Zürich, 1926.**[14]**M. H. Stone,*Irregular differential systems of order two and the related expansion problems*, Trans. Amer. Math. Soc. vol. 29 (1927) pp. 23-53. MR**1501375****[15]**-,*Linear transformations in Hilbert space and their applications to analysis*, Amer. Math. Soc. Colloquium Publications, vol. 15, 1932.**[16]**L. E. Ward,*Some third-order irregular boundary value problems*, Trans. Amer. Math. Soc. vol. 29 (1927) pp. 716-745. MR**1501411**

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DOI:
https://doi.org/10.1090/S0002-9947-1961-0130420-4

Article copyright:
© Copyright 1961
American Mathematical Society