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Transactions of the American Mathematical Society

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Analytic functionals with unbounded carriers and mean periodic functions


Author: Alex Meril
Journal: Trans. Amer. Math. Soc. 278 (1983), 115-136
MSC: Primary 46F15; Secondary 30H05, 32A07, 32A10
DOI: https://doi.org/10.1090/S0002-9947-1983-0697064-1
MathSciNet review: 697064
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Abstract: We study certain ideals in some spaces of analytic functionals with unbounded carriers introduced by T. Kawaï, M. Morimoto and J. W. de Roever. Using Banach algebra methods, we show an example of space without spectral synthesis. Using Hörmander's $ {L^2}$ estimates, we prove a spectral synthesis theorem for mean periodic functions.


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  • [1] E. Amar, $ \bar \partial $ cohomologie $ {C^\infty }$ et applications, Prep. Univ. Paris Sud 80 T. 1.
  • [2] Carlos A. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math. 33 (1979), no. 2, 109–143. MR 544846, https://doi.org/10.1016/S0001-8708(79)80002-X
  • [3] Leon Ehrenpreis, Appendix to the paper “Mean periodic functions I”, Amer. J. Math. 77 (1955), 731–733. MR 0076303, https://doi.org/10.2307/2372594
  • [4] R. Gay, Division des fonctionelles analytiques et fonctions entières de type exponentiel de plusieurs variables, Th. Sc. Math., Strasbourg, 1976.
  • [5] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1962. MR 0133008
  • [6] Lars Hörmander, 𝐿² estimates and existence theorems for the ∂ operator, Acta Math. 113 (1965), 89–152. MR 0179443, https://doi.org/10.1007/BF02391775
  • [7] -, An introduction to complex analysis in several variables, 2nd ed., North-Holland, Amsterdam, 1973.
  • [8] Takahiro Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo Sect. I A Math. 17 (1970), 467–517. MR 0298200
  • [9] Hikosaburo Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366–383. MR 0217557, https://doi.org/10.2969/jmsj/01930366
  • [10] Gottfried Köthe, Topological vector spaces. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Science], vol. 237, Springer-Verlag, New York-Berlin, 1979. MR 551623
  • [11] Pierre Lelong, Fonctionnelles analytiques et fonctions entières (𝑛 variables), Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). Séminaire de Mathématiques Supérieures, No. 13 (Été, 1967). MR 0466606
  • [12] Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 271–355 (French). MR 0086990
  • [13] Mitsuo Morimoto, Analytic functionals with non-compact carrier, Tokyo J. Math. 1 (1978), no. 1, 77–103. MR 502814, https://doi.org/10.3836/tjm/1270216594
  • [14] J. W. de Roever, Fourier transform of holomorphic functions and application to Newton interpolation series. I, Publ. Math. Centrum, Amsterdam, 1974.
  • [15] -, Fourier transform of holomorphic functions and application to Newton interpolation series. II, T. W 148, Math. Centrum, Amsterdam, 1975.
  • [16] J. W. de Roever, Complex Fourier transformation and analytic functionals with unbounded carriers, Mathematical Centre Tracts, vol. 89, Mathematisch Centrum, Amsterdam, 1978. With a preface by E. M. de Jager. MR 494067
  • [17] Yutaka Saburi, Vanishing theorems of cohomology groups with coefficients in sheaves of holomorphic functions with bounds, Proc. Japan Acad. Ser. A Math. Sci. 54 (1978), no. 9, 274–278. MR 517669
  • [18] Laurent Schwartz, Théorie générale des fonctions moyenne-périodiques, Ann. of Math. (2) 48 (1947), 857–929 (French). MR 0023948, https://doi.org/10.2307/1969386
  • [19] François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
  • [20] V. V. Zarinov, Laplace transformation of Fourier hyperfunctions and related classes of analytic functionals. I, Theoret. and Math. Phys. 33 (1978), 1027-1039.
  • [21] -, Laplace transformation of Fourier hyperfunctions and related classes of analytic functionals. II, Theoret. and Math. Phys. 37 (1979), 843-855.

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DOI: https://doi.org/10.1090/S0002-9947-1983-0697064-1
Article copyright: © Copyright 1983 American Mathematical Society