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1994 | 110 | 1 | 65-82
Tytuł artykułu

Continuous linear right inverses for convolution operators in spaces of real analytic functions

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EN
Abstrakty
EN
We determine the convolution operators $T_μ := μ*$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).
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Twórcy
  • Universität Wuppertal, FB 7 - Mathematik Gaussstr. 20, D-42097 Wuppertal, Germany
Bibliografia
  • [1] C. A. Berenstein and B. A. Taylor, A new look at interpolation theory for entire functions of one variable, Adv. in Math. 33 (1979), 109-143.
  • [2] C. A. Berenstein and B. A. Taylor, Interpolation problems in $ℂ^n$ with application to harmonic analysis, J. Analyse Math. 38 (1980), 188-254.
  • [3] R. Braun, R. Meise and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultradifferentiable functions, Proc. London Math. Soc. 61 (1990), 344-370.
  • [4] L. Ehrenpreis, Solution of some problems of division. Part IV. Invertible and elliptic operators, Amer. J. Math. 82 (1960), 522-588.
  • [5] L. Ehrenpreis, Solution of some problems of division. Part V. Hyperbolic operators, ibid. 84 (1962), 324-348.
  • [6] L. Hörmander, An Introduction to Complex Analysis in Several Variables, Princeton Univ. Press, 1967.
  • [7] T. Kawai, On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients, J. Fac. Sci. Univ. Tokyo 17 (1970), 467-517.
  • [8] Yu. F. Korobeĭnik and S. N. Melikhov, A continuous linear right inverse for a representation operator, and conformal mappings, Russian Acad. Sci. Dokl. Math. 45 (1992), 428-431.
  • [9] S. Lang, Complex Analysis, Springer, New York, 1985.
  • [10] M. Langenbruch and S. Momm, Complemented submodules in weighted spaces of analytic functions, Math. Nachr. 157 (1992), 263-276.
  • [11] B. Ya. Levin, Distribution of Zeros of Entire Functions, Amer. Math. Soc., Providence, R.I., 1980.
  • [12] R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, J. Reine Angew. Math. 363 (1985), 59-95.
  • [13] R. Meise, Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type, Studia Math. 92 (1989), 211-230.
  • [14] R. Meise, B. A. Taylor and D. Vogt, Equivalence of slowly decreasing conditions and local Fourier expansions, Indiana Univ. Math. J. 36 (1987), 729-756.
  • [15] R. Meise and D. Vogt, Characterization of convolution operators on spaces of $C^∞$-functions admitting a continuous linear right inverse, Math. Ann. 279 (1987), 141-155.
  • [16] R. Meise and D. Vogt, Einführung in die Funktionalanalysis, Braunschweig, Wiesbaden, 1992.
  • [17] T. Meyer, Die Fourier-Laplace-Transformation quasianalytischer Funktionale und ihre Anwendung auf Faltungsoperatoren, Diplomarbeit, Düsseldorf, 1989.
  • [18] T. Meyer, Surjektivität von Faltungsoperatoren auf Räumen ultradifferenzierbarer Funktionen vom Roumieu Typ, Dissertation, Düsseldorf, 1992.
  • [19] S. Momm, Partial differential operators of infinite order with constant coefficients on the space of analytic functions on the polydisc, Studia Math. 96 (1990), 51-71.
  • [20] S. Momm, Closed principal ideals in nonradial Hörmander algebras, Arch. Math. (Basel) 58 (1992), 47-55.
  • [21] S. Momm, Convex univalent functions and continuous linear right inverses, J. Funct. Anal. 103 (1992), 85-103.
  • [22] S. Momm, Convolution equations on the analytic functions on convex domains in the plane, Bull. Sci. Math., to appear.
  • [23] M. Poppenberg and D. Vogt, A tame splitting theorem for exact sequences of Fréchet spaces, Math. Z., to appear.
  • [24] K. Schwerdtfeger, Faltungsoperatoren auf Räumen holomorpher und beliebig oft differenzierbarer Funktionen, Dissertation, Düsseldorf, 1982.
  • [25] B. A. Taylor, Linear extension operators for entire functions, Michigan Math. J. 29 (1982), 185-197.
  • [26] D. Vogt, Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen, Manuscripta Math. 37 (1982), 269-301.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv110i1p65bwm
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