Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Meyer, Thomas

Studia Mathematica, Tome 122 (1997), p. 101-129 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $ε_{{ω}}(I)$ denote the space of all ω-ultradifferentiable functions of Roumieu type on an open interval I in ℝ. In the special case ω(t) = t we get the real-analytic functions on I. For $μ ∈ ε_{{ω}}(I)^{\prime }$ with $supp(μ) = {0}$ one can define the convolution operator $T_μ: ε_{{ω}}(I) → ε_{{ω}}(I)$, $T_μ(f)(x):= ⟨μ,f(x-·)⟩$. We give a characterization of the surjectivity of $T_μ$ for quasianalytic classes $ε_{{ω}}(I)$, where I = ℝ or I is an open, bounded interval in ℝ. This characterization is given in terms of the distribution of zeros of the Fourier Laplace transform $\widehat{μ}$ of μ.

Détail

Publié le : 1997-01-01

Zbl 0897.46023

EUDML-ID : urn:eudml:doc:216426

Bibliographie

[00000] [1] L. V. Ahlfors, Conformal Invariants. Topics in Geometric Function Theory, McGraw-Hill, 1973.

[00001] [2] 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. | Zbl 0432.30028

[00002] [3] A. Beurling, Quasianalyticity and general distributions, Lectures 4 and 5, Amer. Math. Soc. Summer Institute (Stanford, 1961).

[00003] [4] G. Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1965), 351-407.

[00004] [5] R. W. Braun, A sufficient criterion for the vanishing of $proj^1$ for (DFS)-spectra, preprint.

[00005] [6] R. W. Braun, R. Meise and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), 206-237. | Zbl 0735.46022

[00006] [7] R. W. Braun, R. Meise and D. Vogt, Existence of fundamental solutions and surjectivity of convolution operators on classes of ultra-differentiable functions, Proc. London Math. Soc. (3) 61 (1990), 344-370. | Zbl 0699.46021

[00007] [8] R. W. Braun, R. Meise and D. Vogt, Characterization of the linear partial differential operators with constant coefficients which are surjective on non-quasianalytic classes of Roumieu type, Math. Nachr. 168 (1994), 19-54. | Zbl 0848.35023

[00008] [9] Yu. F. Korobeĭnik, Solvability of a convolution equation in some classes of analytic functions, Mat. Zametki 49 (2) (1991), 76-83 (in Russian); English transl.: Math. Notes 49 (1991), 165-172.

[00009] [10] G. Köthe, Topological Vector Spaces II, Grundlehren Math. Wiss. 237, Springer, 1979. | Zbl 0417.46001

[00010] [11] M. Langenbruch, Hyperfunction fundamental solutions of surjective convolution operators on real analytic functions, J. Funct. Anal. 131 (1995), 78-93. | Zbl 0841.46025

[00011] [12] R. Meise, Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type, Studia Math. 92 (1989), 211-230. | Zbl 0702.46024

[00012] [13] R. Meise, Sequence space representations for (DFN)-algebras of entire functions modulo closed ideals, J. Reine Angew. Math. 363 (1985), 59-95. | Zbl 0574.46043

[00013] [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. | Zbl 0637.46037

[00014] [15] R. Meise und D. Vogt, Einführung in die Funktionalanalysis, Vieweg, Braunschweig, 1992.

[00015] [16] T. Meyer, Die Fourier-Laplace-Transformation quasianalytischer Funktionale und ihre Anwendung auf Faltungsoperatoren, Diplomarbeit, Düsseldorf, 1989.

[00016] [17] T. Meyer, Surjektivität von Faltungsoperatoren auf Räumen ultradifferenzierbarer Funktionen vom Roumieu Typ, Thesis, Düsseldorf, 1992.

[00017] [18] S. Momm, Closed principal ideals in nonradial Hörmander algebras, Arch. Math. (Basel) 58 (1992), 47-55. | Zbl 0804.46066

[00018] [19] S. Momm, Convolution equations on the analytic functions on convex domains in the plane, Bull. Sci. Math. 118 (1994), 259-270. | Zbl 0819.46039

[00019] [20] S. Momm, Ideale in gewichteten Algebren holomorpher Funktionen auf dem Einheitskreis, Dissertation, Düsseldorf, 1988. | Zbl 0657.32009

[00020] [21] V. V. Napalkov and I. A. Rudakov, Convolution operator in the space of real analytic functions, Mat. Zametki 49 (3) (1991), 57-65 (in Russian); English transl.: Math. Notes 49 (1991), 266-271. | Zbl 0763.47012

[00021] [22] V. P. Palamodov, The projective limit functor in the category of linear topological spaces, Math. USSR-Sb. 4 (1968), 529-559. | Zbl 0175.41801

[00022] [23] C. Roumieu, Sur quelques extensions de la notion de distribution, Ann. Sci. École Norm. Sup. (3) 77 (1960), 41-121. | Zbl 0104.33403

[00023] [24] B. A. Taylor, Analytically uniform spaces of infinitely differentiable functions, Comm. Pure Appl. Math. 24 (1971), 39-51. | Zbl 0205.41501

[00024] [25] D. Vogt, Lectures on projective spectra of (DF)-spaces, Universität Wuppertal, 1987.

[00025] [26] D. Vogt, Topics on projective spectra of (LB)-spaces, in: Advances in the Theory of Fréchet Spaces, T. Terzioğlu (ed.), NATO Adv. Sci. Inst. Ser. C 287, Kluwer, Dordrecht, 1989, 11-27.

[00026] [27] D. Vogt, Regularity properties of (LF)-spaces, in: Progress in Functional Analysis, K. D. Bierstedt, J. Bonet, J. Horváth and M. Maestre (eds.), North-Holland Math. Stud. 170, North-Holland, 1992, 57-84. | Zbl 0779.46005

[00027] [28] J. Wengenroth, Acyclic inductive spectra of Fréchet spaces, Studia Math. 120 (1996), 247-258. | Zbl 0863.46002