Article

Full entry | PDF   (0.2 MB)
Keywords:
Banach spaces; holomorphy types; homogeneous polynomials; holomorphic functions; convolution operators; Borel transform; approximation and existence theorems
Summary:
In this paper spaces of entire functions of $\Theta$-holomorphy type of bounded type are introduced and results involving these spaces are proved. In particular, we construct an algorithm'' to obtain a duality result via the Borel transform and to prove existence and approximation results for convolution equations. The results we prove generalize previous results of this type due to B. Malgrange: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales de l'Institute Fourier (Grenoble) VI, 1955/56, 271--355; C. Gupta: Convolution Operators and Holomorphic Mappings on a Banach Space, Séminaire d'Analyse Moderne, 2, Université de Sherbrooke, Sherbrooke, 1969; M. Matos: Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations, IMECC-UNICAMP, 2007; and X. Mujica: Aplicações $\tau (p;q)$-somantes e $\sigma (p)$-nucleares, Thesis, Universidade Estadual de Campinas, 2006.
References:
[1] Banach, S.: Théorie des opérations linéaires. Hafner New York (1932). Zbl 0005.20901
[2] Dineen, S.: Holomorphy types on a Banach space. Stud. Math. 39 (1971), 241-288. MR 0304705 | Zbl 0235.32013
[3] Fávaro, V. V.: The Fourier-Borel transform between spaces of entire functions of a given type and order. Port. Math. 65 (2008), 285-309. DOI 10.4171/PM/1813 | MR 2428422
[4] Fávaro, V. V.: Convolution equations on spaces of quasi-nuclear functions of a given type and order. Preprint.
[5] Floret, K.: Natural norms on symmetric tensor products of normed spaces. Note Mat. 17 (1997), 153-188. MR 1749787 | Zbl 0961.46013
[6] Gupta, C.: Convolution Operators and Holomorphic Mappings on a Banach Space. Séminaire d'Analyse Moderne, 2. Université de Sherbrooke Sherbrooke (1969).
[7] Horváth, J.: Topological Vector Spaces and Distribuitions. Addison-Wesley Reading (1966). MR 0205028
[8] Malgrange, B.: Existence et approximation des équations aux dérivées partielles et des équations des convolutions. Annales de l'Institute Fourier (Grenoble) VI (1955/56), 271-355. MR 0086990
[9] Martineau, A.: Équations différentielles d'ordre infini. Bull. Soc. Math. Fr. 95 (1967), 109-154 French. MR 1507968 | Zbl 0167.44202
[10] Matos, M. C.: On the Fourier-Borel transformation and spaces of entire functions in a normed space. In: Functional Analysis, Holomorphy and Approximation Theory II. North-Holland Math. Studies. G. I. Zapata North-Holland Amsterdam (1984), 139-170. DOI 10.1016/S0304-0208(08)70827-2 | MR 0771327 | Zbl 0568.46036
[11] Matos, M. C.: On convolution operators in spaces of entire functions of a given type and order. In: Complex Analysis, Functional Analysis and Approximation Theory J. Mujica North-Holland Math. Studies Vol. 125 North-Holland Amsterdam (1986), 129-171. DOI 10.1016/S0304-0208(08)72168-6 | MR 0893415 | Zbl 0658.46016
[12] Matos, M. C.: Absolutely Summing Mappings, Nuclear Mappings and Convolution Equations. IMECC-UNICAMP (2007),\hfil http://www.ime.unicamp.br/rel\_pesq/2007/rp03-07.html
[13] Mujica, X.: Aplicações $\tau(p;q)$-somantes e $\sigma(p)$-nucleares. Thesis Universidade Estadual de Campinas (2006).
[14] Nachbin, L.: Topology on Spaces of Holomorphic Mappings. Springer New York (1969). MR 0254579 | Zbl 0172.39902
[15] Pietsch, A.: Ideals of multilinear functionals. In: Proc. 2nd Int. Conf. Operator Algebras, Ideals and Their Applications in Theoretical Physics, Leipzin 1983 Teubner Leipzig (1984), 185-199. MR 0763541 | Zbl 0562.47037
[16] Pietsch, A.: Ideals of multilinear functionals. In: Proc. 2nd Int. Conf. Operator Algebras, Ideals and Their Applications in Theoretical Physics, Leipzin 1983 Teubner Leipzig (1984), 185-199. MR 0763541 | Zbl 0562.47037

Partner of