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On the Borel mapping in the quasianalytic setting

Armin Rainer, Gerhard Schindl

Abstract


The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the real analytic class is never onto the corresponding sequence space.


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References


Bang, T., The theory of metric spaces applied to infinitely differentiable functions, Math. Scand. 1 (1953), 137–152.

Bernstein, S., Sur la définition et les propriétés des fonctions analytiques d'une variable réelle, Math. Ann. 75 (1914), no. 4, 449–468. https://doi.org/10.1007/BF01563654

Beurling, A. K.-A., Quasi-analyticity and general distributions, Lecture notes (mimeographed), AMS Summer Institute, Stanford, 1961.

Björck, G., Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351–407. https://doi.org/10.1007/BF02590963

Boas, Jr., R. P., Signs of derivatives and analytic behavior, Amer. Math. Monthly 78 (1971), 1085–1093. https://doi.org/10.2307/2316310

Bonet, J. and Meise, R., On the theorem of Borel for quasianalytic classes, Math. Scand. 112 (2013), no. 2, 302–319. https://doi.org/10.7146/math.scand.a-15246

Bonet, J., Meise, R., and Melikhov, S. N., A comparison of two different ways to define classes of ultradifferentiable functions, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 425–444.

Braun, R. W., Meise, R., and Taylor, B. A., Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3-4, 206–237. https://doi.org/10.1007/BF03322459

Bruna, J., On inverse-closed algebras of infinitely differentiable functions, Studia Math. 69 (1980/81), no. 1, 59–68.

Carleman, T., Sur le calcul effectif d'une fonction quasi analytique dont on donne les dérivées en un point, C. R. Acad. Sci., Paris 176 (1923), 64–65.

Carleman, T., Les fonctions quasi analytiques, Cellection Borel, Gauthier-Villars, Paris, 1926.

Kriegl, A., Michor, P. W., and Rainer, A., The convenient setting for Denjoy-Carleman differentiable mappings of Beurling and Roumieu type, Rev. Mat. Complut. 28 (2015), no. 3, 549–597. https://doi.org/10.1007/s13163-014-0167-1

Nazarov, F., Sodin, M., and Volberg, A., Lower bounds for quasianalytic functions. I. How to control smooth functions, Math. Scand. 95 (2004), no. 1, 59–79. https://doi.org/10.7146/math.scand.a-14449

Rainer, A. and Schindl, G., Composition in ultradifferentiable classes, Studia Math. 224 (2014), no. 2, 97–131. https://doi.org/10.4064/sm224-2-1

Rainer, A. and Schindl, G., Equivalence of stability properties for ultradifferentiable function classes, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 110 (2016), no. 1, 17–32. https://doi.org/10.1007/s13398-014-0216-0

Schindl, G., Exponential laws for classes of Denjoy-Carleman differentiable mappings, Ph.D. thesis, Universität Wien, 2014.

Schindl, G., Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier transform, Note Mat. 36 (2016), no. 2, 1-35. https://doi.org/10.1285/i15900932v36n2p1

Sfouli, H., On a problem concerning quasianalytic local rings, Ann. Polon. Math. 111 (2014), no. 1, 13–20. https://doi.org/10.4064/ap111-1-2

Thilliez, V., On quasianalytic local rings, Expo. Math. 26 (2008), no. 1, 1–23. https://doi.org/10.1016/j.exmath.2007.04.001

Widder, D. V., The Laplace Transform, Princeton Mathematical Series, v. 6, Princeton University Press, Princeton, N. J., 1941.




DOI: http://dx.doi.org/10.7146/math.scand.a-97101

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