Continuation of holomorphic solutions to convolution equations in complex domains

• Autorzy: Ryuichi Ishimura , Jun-ichi Okada , Yasunori Okada
• Języki publikacji: EN

Abstrakty

EN

For an analytic functional $S$ on $ℂ^n$, we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in $ℂ^n$. We determine the directions in which every solution can be continued analytically, by using the characteristic set.

Słowa kluczowe

EN

convolution equation; analytic continuation; characteristic set

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2000
• Tom: 74
• Numer: 1
• Strony: 105-115

Daty

wydano

2000

otrzymano

1998-12-08

poprawiono

1999-07-28

poprawiono

2000-09-05

Twórcy

autor

Ryuichi Ishimura

• Department of Mathematics and Informatics, Faculty of Sciences, Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

autor

Jun-ichi Okada

• Institute of Natural Sciences, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

autor

Yasunori Okada

• Department of Mathematics and Informatics, Faculty of Sciences, Chiba University, Yayoi-cho, Inage-ku, Chiba 263-8522, Japan

Bibliografia

• [A] T. Aoki, Existence and continuation of holomorphic solutions of differential equations of infinite order, Adv. in Math. 72 (1988), 261-283.
• [I1] R. Ishimura, A remark on the characteristic set for convolution equations, Mem. Fac. Sci. Kyushu Univ. 46 (1992), 195-199.
• [I2] R. Ishimura, The characteristic set for differential-difference equations in real domains, Kyushu J. Math. 53 (1999), 107-114.
• [I-O1] R. Ishimura and Y. Okada, The existence and the continuation of holomorphic solutions for convolution equations in tube domains, Bull. Soc. Math. France 122 (1994), 413-433.
• [I-O2] R. Ishimura and Y. Okada, The micro-support of the complex defined by a convolution operator in tube domains, in: Singularities and Differential Equations, Banach Center Publ. 33, Inst. Math., Polish Acad. Sci., 1996, 105-114.
• [I-O3] R. Ishimura and Y. Okada, Examples of convolution operators with described characteristics, in preparation.
• [I-Oj] R. Ishimura and Y. Okada, Sur la condition (S) de Kawai et la propriété de croissance régulière d'une fonction sous-harmonique et d'une fonction entière, Kyushu J. Math. 48 (1994), 257-263.
• [Ka] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 17 (1970), 467-517.
• [Ki] C. O. Kiselman, Prolongement des solutions d'une équation aux dérivées partielles à coefficients constants, Bull. Soc. Math. France 97 (1969), 329-356.
• [Ko] Yu. F. Korobeĭnik, Convolution equations in the complex domain, Math. USSR-Sb. 36 (1985), 171-193.
• [Kr] A. S. Krivosheev, A criterion for the solvability of nonhomogeneous convolution equations in convex domains of $ℂ^n$, Math. USSR-Izv. 36 (1991), 497-517.
• [Ll-G] P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Grundlehren Math. Wiss. 282, Springer, Berlin, 1986.
• [Lv] B. Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monographs, Amer. Math. Soc., Providence, 1964.
• [R] L. I. Ronkin, Functions of Completely Regular Growth, Kluwer, 1992.
• [Sé] A. Sébbar, Prolongement des solutions holomorphes de certains opérateurs différentiels d'ordre infini à coefficients constants, in: Séminaire Lelong-Skoda, Lecture Notes in Math. 822, Springer, Berlin, 1980, 199-220.
• [V] A. Vidras, Interpolation and division problems in spaces of entire functions with growth conditions and their applications, Doct. Diss., Univ. of Maryland.
• [Z] M. Zerner, Domaines d'holomorphie des fonctions vérifiant une équation aux dérivées partielles, C. R. Acad. Sci. Paris 272 (1971), 1646-1648.