A~Paley--Wiener theorem for generalized entire functions on infinite-dimensional spaces

A. Yu. Khrennikov ; H. Petersson

Izvestiya. Mathematics , Tome 65 (2001) no. 2, pp. 403-424.

Résumé

We study entire functions on infinite-dimensional spaces. The basis is the study of spaces of Gateaux holomorphic functions that are bounded on certain subsets (bounded entire functions). The main goal is to characterize the Fourier image of the corresponding spaces of generalized entire functions (ultra-distributions) by an infinite-dimensional Paley–Wiener theorem. We introduce entire functions of exponential type and prove a generalization of the classical Paley–Wiener theorem. The crucial point of our theory is the dimension-invariant estimate given by Lemma 4.12.

Bibliographie

[1] Fomin S. V., “Obobschennye funktsii beskonechnogo chisla peremennykh i ikh preobrazovanie Fure”, UMN, 23:2 (1968), 215–216

[2] Smolyanov O. G., “Beskonechnomernye psevdodifferentsialnye operatory i kvantovanie po Shredingeru”, DAN SSSR, 263 (1982), 558–562 | MR | Zbl

[3] Bogachev V. I., Smolyanov O. G., “Analiticheskie svoistva beskonechnomernykh raspredelenii”, UMN, 45:3 (1990), 3–84 | MR

[4] Smolyanov O. G., Khrennikov A. Yu., “Tsentralnaya predelnaya teorema dlya obobschennykh mer na beskonechnomernom prostranstve”, DAN SSSR, 281 (1985), 279–283 | MR | Zbl

[5] Smolyanov O. G., Khrennikov A. Yu., “Algebra beskonechnomernykh psevdodifferentsialnykh operatorov”, DAN SSSR, 292:6 (1987), 1310–1314 | MR

[6] Smolyanov O. G., Shavgulidze E. T., Kontinualnye integraly, Izd-vo MGU, M., 1990 | MR | Zbl

[7] Uglanov A. V., “Ob odnoi konstruktsii feinmanovskogo integrala”, DAN SSSR, 262 (1982), 37–40 | MR | Zbl

[8] Khrennikov A. Yu., “Beskonechnomernye psevdodifferentsialnye operatory i ikh prilozheniya k kvantovoi teorii polya”, Izv. AN SSSR. Ser. matem., 51:6 (1987), 1265–1291 | MR

[9] Alberverio S., Hoegh-Krohn R., Mathematical theory of Feynman path integral, Lecture Notes in Math., 523, 1976

[10] Kondratiev Yu., Streit L., Westerkamp W., Yan J.-A., Generalized functions in an infinite dimensional analysis, IIAS Report No 1995-002, 1995

[11] Berezanskii Yu. M., Kondratev Yu. G., Spektralnye metody v beskonechnomernom analize, Nauk. dumka, Kiev, 1988 | MR | Zbl

[12] Khille E., Funktsionalnyi analiz i polugruppy, IL, M., 1951

[13] Dineen S., Complex analysis in locally convex spaces, North Holland, Amsterdam, 1981 | MR | Zbl

[14] Dineen S., Complex analysis on infinite dimensional spaces, Springer-Verlag, Berlin a.o., 1999 | MR

[15] Dineen S., “Unbounded holomorphic functions on a Banach space”, J. London Math. Soc., 2 (1972), 461–465 | DOI | MR

[16] Nachbin L., Topology on spaces of holomorphic mappings, Springer-Verlag, Berlin, 1969 | MR | Zbl

[17] Aron R. M., “Entire functions of unbounded type on a Banach space”, Boll. Un. Mat. Ital., 9 (1974), 28–31 | MR | Zbl

[18] Kiselman C. O., “Geometric aspects of the theory of bounds for entire functions in normed spaces”, Infinite Dimensional Holomorhpy and Applications, North Holland Mathematics Studies, 12, ed. M. C. Matos, 1977, 249–275 | MR | Zbl

[19] Kiselman C. O., “On the radius of convergence of an entire function in a normed space”, Ann. Polon. Math., 33 (1976), 39–55 | MR | Zbl

[20] Kiselman C. O., Croissance des fonctions plurisousharmoniques en dimension infinie, Preprint Univ. Upsalla, 1984 | MR

[21] Kiselman C. O., “The partial Legendre transformation for plurisubharmonic functions”, Invent. Math., 49 (1978), 137–148 | DOI | MR | Zbl

[22] Kiselman C. O., “Plurisubharmonic functions and plurisubharmonic topologies”, Advances in holomorphy, 1979, 431–449 | MR | Zbl

[23] Lempert L., “The Dolbeault complex in infinite dimensions, 1”, J. of the Amer. Math. Soc., 11 (1998), 485–520 | DOI | MR | Zbl

[24] Lempert L., “The Dolbeault complex in infinite dimensions, 2”, J. of the Amer. Math. Soc., 12 (1999), 775–793 | DOI | MR | Zbl

[25] Khrennikov A. Yu., “Feinmanovskie integraly na fazovom prostranstve i simvoly beskonechnomernykh psevdodifferentsialnykh operatorov”, Matem. zametki, 37:5 (1985), 734–742 | MR

[26] Khrennikov A. Yu., “Vtorichnoe kvantovanie i psevdodifferentsialnye operatory”, TMF, 66 (1986), 339–349 | MR | Zbl

[27] Khrennikov A., $p$-adic valued distributions in mathematical physics, Kluwer Acad, Dordrecht, 1994 | MR | Zbl

[28] Khrennikov A., Non-Archimedian Analysis: Quantum Paradoxes, Dynamical systems and Biological Models, Kluwer Acad, Dordrecht, 1997 | MR | Zbl

[29] Khrennikov A. Yu., “Obobschennye funktsii i gaussovskie kontinualnye integraly po nearkhimedovym funktsionalnym prostranstvam”, Izv. RAN. Ser. matem., 55:4 (1991), 780–814 | MR

[30] Khrennikov A. Yu., “Integrirovanie po obobschennym meram na topologicheskikh vektornykh prostranstvakh”, Tr. Mosk. matem. ob-va, 49 (1986), 115–132 | MR

[31] Albeverio S., Khrennikov A., Smolyanov O., “The probabilistic Feynman–Kac formula for an infinite-dimensional Schrödinger equation with exponential and singular potentials”, Potential Analysis, 11 (1999), 157–181 | DOI | MR | Zbl

[32] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[33] Hida T., “Analysis of Brownian functionals”, Carleton Math. Lecture notes, 13, Carleton Univ., 1975 | MR | Zbl

[34] Hida T., Kuo H.-H., Potthoff J., Streit L., White noise: an infinite dimensional calculus, Kluwer Academic, Dordrecht, 1993 | MR | Zbl

[35] Kuo H.-H., White noise distribution theory, CRC Press, Boca-Raton, 1996 | MR | Zbl

[36] Pich A., Yadernye lokalno vypuklye prostranstva, Mir, M., 1967 | MR

[37] Horvath J., Topological vector spaces and distributions. I. Reading, Addison-Wesley, Mass., 1966 | MR | Zbl

[38] Ahlfors L., Complex analysis, 3d ed., McGraw-Hill, N.Y., 1979 | MR | Zbl

[39] Treves F., Linear partial differential equations with constant coefficients, Gordon and Breach, N. Y., 1966 | MR | Zbl

[40] Petersson H., Properties of analytic maps on locally convex spaces, Reports from Växjö University – Mathematics, natural sciences and technology, No 1, 1999, ISBN 91-7636-206-X

[41] Petersson H., Generalized functions on $\mathbb C^\infty $ and $\bigoplus _k^\infty \mathbb C_k$, Reports from Växjö University–Mathematics, natural sciences and technology, No 4, 1999, ISBN 91-7636-210-8

[42] Petersson H., Bounded entire functions on infinite-dimensional spaces and Paley–Wiener theorem, Reports from Växjö University–Mathematics, natural sciences and technology, No 9, 1999, ISBN 91-7636-216-7

[43] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki. T. 1. Funktsionalnyi analiz, Mir, M., 1977 | MR