Complex Analysis/Functional Analysis
Expansion in series of exponential polynomials of mean-periodic functions with growth conditions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 509-514.

Let θ be a Young function. Consider the space Fθ(C) of all entire functions on C with θ-exponential growth. In this Note, we are interested in the solutions fFθ(C) of the convolution equation Tf=0, called T-mean-periodic functions, where T is in the topological dual of Fθ(C). We show that each mean-periodic function admits an expansion as a convergent series of exponential polynomials.

Soit θ une fonction de Young. Considérons l'espace Fθ(C) de toutes les fonctions entières sur C à croissance θ-exponentielle. On s'intéresse dans cette Note aux solutions fFθ(C) de l'équation de convolution Tf=0, appelées fonctions T-moyenne-périodiques, où T est dans le dual topologique de Fθ(C). On montre que toute fonction moyenne-périodique admet un développement en série de polynômes exponentiels. De plus cette série est convergente pour la topologie de l'espace Fθ(C).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.03.027
Ouerdiane, Habib 1; Ounaies, Myriam 2

1 Département de mathématique, faculté des sciences de Tunis, Université de Tunis El Manar, campus universitaire, 1060 Tunis, Tunisia
2 Institut de recherche mathématique avancée, Université Louis-Pasteur, 7, rue René-Descartes, 67084 Strasbourg cedex, France
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Ouerdiane, Habib; Ounaies, Myriam. Expansion in series of exponential polynomials of mean-periodic functions with growth conditions. Comptes Rendus. Mathématique, Volume 346 (2008) no. 9-10, pp. 509-514. doi : 10.1016/j.crma.2008.03.027. http://www.numdam.org/articles/10.1016/j.crma.2008.03.027/

[1] Berenstein, C.A.; Gay, R. Complex Analysis and Special Topics in Harmonic Analysis, Springer-Verlag, New York, 1995

[2] Berenstein, C.A.; Struppa, D.C. Dirichlet series and convolution equations, Publ. RIMS, Kyoto Univ., Volume 24 (1988), pp. 783-810

[3] Berenstein, C.A.; Struppa, D.C. Complex analysis and convolution equations, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya., vol. 54, 1989, pp. 5-111 (English transl.: Several Complex Variables. V: Complex Analysis in Partial Differential Equations and Mathematical Physics, Encycl. Math. Sci., vol. 54, 1993, pp. 1-108)

[4] Berenstein, C.A.; Taylor, B.A. A new look at interpolation theory for entire functions of one variable, Adv. in Math., Volume 33 (1979) no. 2, pp. 109-143

[5] Gannoun, R.; Hachaichi, R.; Ouerdiane, H.; Rezgui, A. Un théorème de dualité entre espaces de fonctions holomorphes à croissance exponentielle, J. Funct. Anal., Volume 171 (2000) no. 1, pp. 1-14

[6] Isaacson, E.; Keller, H.B. Analysis of Numerical Methods, Dover Publications Inc., New York, 1994 (Corrected reprint of the 1966 original, Wiley, New York)

[7] Ounaïes, M. Interpolation by entire functions with growth conditions, Michigan Math. J., Volume 56 (2008) no. 1

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