Hostname: page-component-848d4c4894-2pzkn Total loading time: 0 Render date: 2024-05-08T20:34:52.461Z Has data issue: false hasContentIssue false
  • English
  • Français

Strongly irregular boundary value problems

Published online by Cambridge University Press:  14 November 2011

Bernd Schultze
Bernd Schultze
Affiliation:
Universität Essen, Fachbereich 6, Mathematik, 43 Essen 1, Germany
Get access Rights & Permissions [Opens in a new window]

Synopsis

A new class of irregular boundary value problems—non-regular in the sense of Birkhoff—is studied. This class of strongly irregular problems includes the class of boundary value problems with irregular decomposing boundary conditions. For each strongly irregular problem we can find a problem with irregular decomposing boundary conditions so that we have equiconvergence with respect to Riesz typical means of the eigenfunction expansions arising from these two problems of an arbitrary summable function.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 82 , Issue 3-4 , 1979 , pp. 291 - 303
Copyright
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Benzinger, H. E.. Green′s function for ordinary differential operators. J. Differential Equations 7 (1970), 478–496.Google Scholar
2Birkhoff, G. D.. Boundary value and expansion problems of ordinary linear differential equations. Trans. Amer. Math. Soc. 9 (1908), 373–395.Google Scholar
3Eberhard, W.. Das asymptotische Verhalten der Greenschen Funktion irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen. Math. Z. 86 (1964), 45–53.CrossRefGoogle Scholar
4Eberhard, W.. Die Entwicklungen nach Eigenfunktionen irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen. Math. Z. 86 (1964), 205–214.CrossRefGoogle Scholar
5Eberhard, W.. Die Entwicklungen nach Eigenfunktionen irregulärer Eigenwertprobleme mit zerfallenden Randbedingungen II. Math. Z. 90 (1965), 126–137.CrossRefGoogle Scholar
6Eberhard, W.. Zur Vollstandigkeit des Biorthogonalsystems von Eigenfunktionen irregulärer Eigenwertprobleme. Math. Z. 146 (1976), 213–221.Google Scholar
7Hopkins, J. W.. Some convergent developments associated with irregular boundary conditions. Trans. Amer. Math. Soc. 20 (1919), 245–259.Google Scholar
8Keldysh, M. V.. On eigenvalues and eigenfunctions of some classes of nonselfadjoint equations. Dokl. Akad. Nauk SSSR 77 (1951), 11–14.Google Scholar
9Khromov, A. P.. Eigenfunction expansions for ordinary differential operators in a finite interval. Dokl. Akad. Nauk SSSR 146 (1962), 1294–1297.Google Scholar
10Khromov, A. P.. Eigenfunction expansions for ordinary differential operators with irregular separable boundary conditions. Dokl. Akad. Nauk SSSR 152 (1963), 1324–1326.Google Scholar
11Langer, R. E.. The zeros of exponential sums and integrals. Bull. Amer. Math. Soc. 37 (1931), 213–239.CrossRefGoogle Scholar
12Naimark, M. A.. Linear differential operators (New York: Ungar; 1967).Google Scholar
13Seifert, G.. A third order irregular boundary value problem and the associated series. Pacific J. Math. 2 (1952), 395–406.Google Scholar
14Stone, M. H.. A comparison of the series of Fourier and Birkhoff. Trans. Amer. Math. Soc. 28 (1926), 695–761.CrossRefGoogle Scholar
15Stone, M. H.. Irregular differential systems of order two and related expansion problems. Trans. Amer. Math. Soc. 29 (1927), 23–53.CrossRefGoogle Scholar
16Ward, L. E.. An irregular boundary value and expansion problem. Ann. of Math. 26 (1925), 21–36.Google Scholar
17Ward, L. E.. A third order irregular boundary value problem and the associated series. Trans. Amer. Math. Soc. 34 (1932), 417–434.Google Scholar