A Left-Definite Study of Legendre’s Differential Equation and of the Fourth-Order Legendre Type Differential Equation

Results in Mathematics

Volumn 37, Issue 1-2, 2000, Pages 155-196

  Vonhoff, Rolf ^a  

^a TU DORTMUND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000875598     PISSN: 14226383     EISSN: 14209012     Source Type: Journal    
DOI: 10.1007/BF03322519     Document Type: Article

References (46)

1. N. I. Achieser and I. M. Glasmann: Theorie der linearen Operatoren im Hilbertraum, 8., erw. Aufl., hg. von H. Baumgärtel, Thun, Frankfurt am Main 1981
2. Christer Bennewitz: A Generalization of Niessen’s Limit-Circle Criterion, Proceedings of the Royal Society of Edinburgh, Vol. 78A (1977), pp. 81–90
3. W. N. Everitt: Legendre Polynomials and Singular Differential Operators, Lecture Notes in Mathematics 827 (1980), pp. 83–106
4. W. N. Everitt, A.M. Krall, and L. L. Littlejohn: On Some Properties of the Legendre Type Differential Expression, Quaestiones Mathematicae, Vol. 13 (1990), pp. 83–106
5. W. N. Everitt, A. M. Krall, L. L. Littlejohn, and V. P. Onyango-Otieno: Differential Operators and the Laguerre Type Polynomials, SIAM Journal on Mathematical Analysis, Vol. 23 (1992), pp. 722–736
6. W. N. Everitt and L. L. Littlejohn: Differential Operators and the Legendre Type Polynomials, Differential and Integral Equations, Vol. 1 (1988), pp. 97–116
7. W. N. Everitt and L. L. Littlejohn: The Density of Polynomials in a Weighted Sobolev Space, Rendiconti di Matematica, Vol. 10 (1990), pp. 835–852
8. W. N. Everitt and L. L. Littlejohn: Orthogonal Polynomials and Spectral Theory: A Survey, Orthogonal Polynomials and their Applications, C. Brezinski, L. Gori and A. Ronveaux (Editors), Proceedings of the third International Symposium held in Erice, Italy 1990 (1991), pp. 21-55
9. (r) Differential Expressions, Research Report 3/92/57, Department of Mathematics and Statistics, Utah State University 1993
10. W. N. Everitt, L. L. Littlejohn, and S. M. Loveland: Some Properties of the Sixth-Order Legendre-Type Differential Expression, Rendiconti di Matematica, Serie VII, Vol. 13 (1993), pp. 773–799
11. W. N. Everitt, L. L. Littlejohn, and S. M. Loveland: Properties of the Operator Domains of the Fourth-Order Legendre-Type Differential Expressions, Differential and Integral Equations, Vol. 7 (1994), pp. 795–810
12. W. N. Everitt, L. L. Littlejohn, and S. C. Williams: Orthogonal Polynomials in Weighted Sobolev Spaces, Lecture Notes in Pure and Applied Mathematics 117, Dekker, New York 1989, pp. 53–72
13. W. N. Everitt, L. L. Littlejohn, and S. C. Williams: The Left-Definite Legendre Type Boundary Problem, Constructive Approximation, Vol. 7 (1991), pp. 485–500
14. W. N. Everitt, L. L. Littlejohn, and S. C. Williams: Orthogonal Polynomials and Approximation in Sobolev Spaces, Journal of Computational and Applied Mathematics, Vol. 48 (1993), pp. 69–90
15. Mojdeh Hajmirzaahmad: Jacobi Polynomial Expansions, Journal of Mathematical Analysis and Applications, Vol. 181 (1994), pp. 35–61
16. E. Kamke: Differentialgleichungen. Lösungsmethoden und Lösungen, I. Gewöhnliche Differentialgleichungen, 8. durchgesehene Auflage, Akademische Verlagsgesellschaft Geest &; Portig K.-G., Leipzig 1967
17. Allan M. Krall: Orthogonal Polynomials Satisfying Fourth Order Differential Equations, Proceedings of the Royal Society of Edinburgh, Section A, Vol. 87 (1981), pp. 271–288
18. Allan M. Krall: A Review of Orthogonal Polynomials Satisfying Boundary Value Problems, Lecture Notes in Mathematics 1329 (1988), pp. 73–97
19. Allan M. Krall, W. N. Everitt, and L. L. Littlejohn: The Legendre Type Operator in a Left Definite Eilbert Space, Quaestiones Mathematicae, Vol. 15 (1992), pp. 467–475
20. Allan M. Krall, W. N. Everitt, L. L. Littlejohn, and V. P. Onyango-Otieno: The Laguerre Type Operator in a Left Definite Hilbert Space, Journal of Mathematical Analysis and Applications, Vol. 192 (1995), pp. 460–468
21. Allan M. Krall and L. L. Littlejohn: The Legendre Polynomials Under a Left Definite Energy Norm, Quaestiones Mathematicae, Vol. 16 (1993), pp. 393–403
22. H. L. Krall: Certain Differential Equations for Tchebyeheff Polynomials, Duke Mathematical Journal, Vol. 4 (1938), pp. 705–718
23. Adrien-Marie Legendre: Suite des Recherches sur la Figure des Planétes, Histoire de l’Académie Royale des Sciences avec les Mémoires de Mathématique et de Physique, Paris 1789 (1793), pp. 372-454
24. Lance L. Littlejohn: The Krall Polynomials: A New Class of Orthogonal Polynomials, Quaestiones Mathematicae, Vol. 5 (1982), pp. 255–265
25. Lance L. Littlejohn and Allan M. Krall: Orthogonal Polynomials and Singular Sturm-Liouville Systems, I, The Rocky Mountain Journal of Mathematics, Vol. 16 (1986), pp. 435–479
26. Lance L. Littlejohn and Allan M. Krall: Orthogonal Polynomials and Higher Order Singular Sturm-Liouville Systems, Acta Applicandae Mathematicae 17 (1989), pp. 99–170
27. Susan Marie Loveland: Spectral Analysis of the Legendre Equations, Dissertation, Utah State University, Logan (1990)
28. Heinz-Dieter Niessen: Zum verallgemeinerten zweiten Weylschen Satz, Archiv der Mathematik, Vol. 22 (1971), S. 648–656
29. Heinz-Dieter Niessen: Verallgemeinerte S-hermitesche Eigenwertprobleme, unpublished manuscript 1973
30. Heinz-Dieter Niessen: A Necessary and Sufficient Limit-Circle Criterion for Left-Definite Eigenvalue Problems, Lecture Notes in Mathematics 415 (1974), pp. 205–210
31. H.-D. Niessen and A. Schneider: Spectral Theory for Left-Definite Singular Systems of Differential Equations I and II, Mathematical Studies 13 (1974), S. 29–56 (= North-Holland Publishing Company)
32. Åke Pleuel: A Survey of Spectral Theory for Pairs of Ordinary Differential Operators, Lecture Notes in Mathematics 448 (1975), S. 256–270
33. Åke Pleuel: On Legendre’s Polynomials, Mathematics Studies 21 (1975), pp. 175–180 (= North Holland Publishing Company)
34. Åke Pleuel: On the Boundary Condition for the Legendre Polynomials, Annales Academiæ Scientiarum Fennicæ, Series A, I. Mathematica, Vol. 2 (1976), pp. 397–408
35. Albert Schneider: Untersuchungen über singuläre reelle S-hermitesche Differentialgleichungssysteme im Normalfall, Mathematische Zeitschrift, Vol. 107 (1968), pp. 271–296
36. Albert Schneider: Geometrische Bedeutung eines Satzes vom Weylschen Typ für Shermitesche Differentialgleichungssysteme im Normalfall, Archiv der Mathematik, Vol. 20 (1969), pp. 147–154
37. Albert Schneider: On Spectral Theory for the Linear Self adjoint Equation Fy = λGy, Lecture Notes in Mathematics 846 (1981), pp. 306–332
38. Albert Schneider and Heinz-Dieter Niessen: Linksdefinite singuläre kanonische Eigenwertprobleme I, Journal für die reine und angewandte Mathematik, Vol. 281 (1976), pp. 13–52
39. Albert Schneider and Heinz-Dieter Niessen: Linksdefinite singuläre kanonische Eigenwertprobleme II, Journal für die reine und angewandte Mathematik, Vol. 289 (1977), pp. 62–84
40. Bernd Schultze: Green’s Matrix and the Formula of Titchmarsh-Kodaira for Singular Left-Definite Canonical Eigenvalue-Problems, Proceedings of the Royal Society of Edinburgh, Vol. 83A (1979), pp. 147–183
41. Gabor Szegö: Orthogonal Polynomials, American Mathematical Society, Colloquium Publications, Volume XXIII, Fourth Edition, Providence, Rhode Island 1978
42. Rolf Vonhoff: Spektraltheoretische Untersuchung linksdefiniter Differentialgleichungen im singulären Fall, Dissertation, Universität Dortmund 1995
43. Rolf Vonhoff: The Sixth-Order Legendre Type Differential Operator in a Weighted Sobolev Space, Preprint 1999 (32 pp.)
44. Rolf Vonhoff: Eigenvalue Problems of Left-Definite Differential Equations, in preparation
45. Hermann Weyl: Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Mathematische Annalen, Vol. 68 (1910), pp. 220–269
46. Hermann Weyl: Über gewöhnliche lineare Differentialgleichungen mit singulären Stellen und ihre Eigenfunktionen (2. Note), Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen. Mathematisch-physikalische Klasse (1910), pp. 442-467