A CHARACTERIZATION OF WEAKLY REGULAR LINEAR FUNCTIONALS
Vol. 31 No. 119 (2007), Mathematics
Vol. 31 No. 119 (2007)

A CHARACTERIZATION OF WEAKLY REGULAR LINEAR FUNCTIONALS

Mathematics
https://doi.org/10.18257/raccefyn.31(119).2007.2335
Published 2023-12-02
Francisco Marcellán
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganés, Spain.
Ridha Sfaxi
Département des Méthodes Quantitatives, Institut Supérieur de Gestion de Gabes. Avenue Jilani Habib 6002, Gabes, Tunisie.
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How to Cite

Marcellán, F., & Sfaxi, R. (2023). A CHARACTERIZATION OF WEAKLY REGULAR LINEAR FUNCTIONALS. Revista De La Academia Colombiana De Ciencias Exactas, Físicas Y Naturales, 31(119), 285–295. https://doi.org/10.18257/raccefyn.31(119).2007.2335
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Abstract

A linear functional is said to be weakly-regular if it is not a finite sum of Dirac masses and their derivatives. In this paper, we consider the first-order linear differential equations (Eu)'+Fu=0 where u is a non-zero linear functional and (E,F) is a pair of polynomials, with E  monic. The aim of this work is to give weak-regularity conditions on u. Under certain admissibility conditions of the pair (E, F), the weak-regularity of u leads to its regularity. Some examples are analyzed.

https://doi.org/10.18257/raccefyn.31(119).2007.2335

Keywords

First-order linear differential equations | weak-regular and regular functionals | weak-semiclassical and semi-classical functionals
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References

M. Bachene, Les polynómes semi-classiques de classe zéro et de classe un. Thèse de troisième cycle, Université Pierre et Marie Curie, Paris, 1986.

S. Belmehdi, Formes linéaires et polynômes orthogonaux semi-classiques de classe s = 1. Description et classification. Thèse d'État, Université Pierre et Marie Curie, Paris, 1990.

S. Bochner, Über Sturm-Liouvillesche Polynomsysteme, Math. Z. 29 (1929), 730-736.

T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 1978.

J. L. Geronimus, On polynomials with respect to numerical sequences and on Hahn's theorem, Izv. Akad. Nauk. 4 (1940), 215-228 (in Russian).

E. Hendriksen & H. Van Rossum, Semiclassical orthogonal polynomials. In Polynômes Orthogonaux et Applications, Bar-le-Duc, 1984. C. Brezinski et al. Editors. Lecture Notes in Math., Vol. 1171, Springer-Verlag, Berlin (1985), 354-361.

P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semiclassiques. In Orthogonal Polynomials and their applications. Proc. Erice. (1990). C. Brezinski et al., Editors. Ann. Comp. Appl. Math., 9 (1991), 95-130.

P. Maroni, Variations around classical orthogonal polynomials. Connected problems. J. Comp. Appl. Math. 48 (1993), 133-155.

P. Maroni, Fonctions eulériennes. Polynômes orthogonaux classiques. Techniques de l'Ingénieur, A 154 (1994), 1-30.

J. Shohat, A differential equation far orthogonal polynomials. Duke Math. J. 5 (1939), 401-417.

V. B. Uvarov, The connection between systems of polynomials orthogonal with respect to different distribution functions. USSR Comput. Math. Phys. (6) 9 (1969), 25-36.

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