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Analytic Extension of Smooth Functions

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Abstract

Let F be a closed proper subset of ℝn and let ℰ* be a class of ultradifferentiable functions. We give a new proof for the following result of Schmets and Valdivia on analytic modification of smooth functions: for every function ƒ ∈ ℰ* (ℝn) there is \({\widetilde f} \in {\cal E}_{*}(\rm R ^{n})\)which is real analytic on ℝnF and such that ∂a ƒ ¦ F = ∂a ƒ ¦ F for any a ∈ ℕ0 n. For bounded ultradifferentiable functions ƒ we can obtain \({\widetilde f}\)by means of a continuous linear operator.

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References

  1. E. F. Beckenbach and R. Bellmann, Inequalities, Springer, Berlin (1983).

    Google Scholar 

  2. A. Beurling, Quasianalyticity and General Distributions, AMS Summer Institute, Lectures 4 and 5, Standford (1961).

  3. G. Björck, “Linear partial differential operators and generalized distributions”, Ark. Mat, Vol. 6 (1965), 351–407.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. Bonet, R. Braun, R. Meise, and B. A. Taylor, “Whitney’s extension theorem for non quasi-analytic functions”, Studia Math. Vol. 99 (1991), 156–184.

    MathSciNet  MATH  Google Scholar 

  5. J. Bonet, R. Meise, and B. A. Taylor, “On the range of the Borel map for classes of quasi-analytic functions”, Progress in Funct Anal., North Holland Math. Studies, Vol. 179 (1991), 97–111.

    MATH  Google Scholar 

  6. M. D. Bronshtein, “Continuation of functions in Carleman’s non quasi-analytic classes”, Sov. Math. (Iz. VUZ), Vol. 30 (1986), 11–14.

    Google Scholar 

  7. R. W. Braun, R. Meise, and B. A. Taylor, “Ultradifferentiable functions and Fourier analysis”, Result. Math., Vol. 17 (1990), 207–237.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Bruna, “An extension theorem of Whitney type for non quasi-analytic classes of functions”, J. London Math. Soc, Vol. 22 (1980), 495–505.

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Chaumat and A.-M. Chollet, “Surjectivité de l’application restriction à un compact dans des classes de fonctions ultradifférentiables”, Math. Ann., Vol. 298 (1994), 7–40.

    Article  MathSciNet  Google Scholar 

  10. U. Pranken, “Continuous linear extension of ultradifferentiable functions of Beurling type”, Math. Nachr., Vol. 164 (1993), 119–139.

    Article  MathSciNet  Google Scholar 

  11. L. Hörmander, The Analysis of Linear Partial Differential Operators I+II, Grundlagen Math. Wiss. Bde. 256+257, Springer, Berlin/Heidelberg/New York (1983).

    Google Scholar 

  12. H. Komatsu, “Ultradistributions I. Structure theorems and a characterization”, J. Fac. Sci. Tokyo Sec. IA, Vol. 20 (1973), 25–105.

    MathSciNet  MATH  Google Scholar 

  13. H. Komatsu, “Ultradistributions II, The kernel theorem and ultradistributions with support in a sub-manifold”, J. Fac. Sci. Univ. Tokyo, Sect. IA, Vol. 24 (1977), 607–628.

    MathSciNet  MATH  Google Scholar 

  14. M. Langenbruch, “Extension of ultradifferentiable functions of Roumieu type”, Archiv der Math., Vol. 51 (1988), 353–362.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Langenbruch, “Differentiable functions and the \(\overline \partial\)-complex”. Functional analysis, Proc. of the Essen Conf., K.D. Bierstedt et al., Dekker Lect. Notes in Pure and Appl. Math., Vol. 150 (1994), 415–434.

    MathSciNet  Google Scholar 

  16. R. Meise and B. A. Taylor, “Whitney’s extension theorem for ultradifferentiable functions of Beurling type”, Ark. Math., Vol. 26 (1988), 265–287.

    Article  MathSciNet  MATH  Google Scholar 

  17. B. S. Mityagin, “An infinitely differentiable function with the values of its derivatives given at a point”, Soviet Math. Dokl, Vol. 2 (1961), 594–597.

    MATH  Google Scholar 

  18. H.-J. Petzsche, “On E. Borel’s Theorem”, Math. Ann., Vol. 282 (1988), 299–313.

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Plesniak, “Extension and polynomial approximation of ultradifferentiable functions in ℝn”, Bull. Soc. Roy. Sc. Liège, Vol. 63 (1994), 393–402.

    MathSciNet  MATH  Google Scholar 

  20. C. Roumieu, “Sur quelques extensions de la notion de distribution”, Ann. École Norm. Sup., Vol. 77 (1960), 41–121.

    MathSciNet  MATH  Google Scholar 

  21. C. Roumieu, “Ultradistributions définies sur ℝn et sur certains classes de variétés differentiables”, J. Anal. Math., Vol. 10 (1962-1963), 153–192.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Schmets, M. Valdivia, “Analytic extension of non quasi-analytic Whitney jets of Roumieu type”, Result. Math., Vol. 31 (1997), 374–385.

    Article  MathSciNet  MATH  Google Scholar 

  23. J. Schmets, M. Valdivia, “Analytic extension of non quasi-analytic Whitney jets of Beurling type”, Math. Nachr., Vol. 195 (1998), 187–197.

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Schmets, M. Valdivia, “Analytic extension of ultradifferentiable Whitney jets”, Collecteana Mathe-matica (to appear).

  25. J. Schmets, M. Valdivia, “On the existence of continuous linear analytic extension maps for Whitney jets”, Bull Polish. Acad. Sc. Math., Vol. 45 (1997), 359–367.

    MathSciNet  MATH  Google Scholar 

  26. R. T. Seeley, “Extension of C∞-functions defined in a halfspace”, Proc. Amer. Math. Soc, Vol. 15 (1964), 625–626.

    MathSciNet  MATH  Google Scholar 

  27. M. Tidten, “Fortsetzungen von C∞-Funktionen, welche auf einer abgeschlossenen Menge in ℝn definiert sind”, Manuscripta Math., Vol. 27 (1979), 291–312.

    Article  MathSciNet  MATH  Google Scholar 

  28. M. Valdivia, “On certain analytic function ranged linear operators in spaces of ultradifferentiable functions”, Mathematica Japonica, Vol. 44 (1996), 415–436.

    MathSciNet  MATH  Google Scholar 

  29. M. Valdivia, “On certain linear operators in spaces of ultradifferentiable functions”, Result. Math., Vol. 36 (1996), 321–345.

    Article  MathSciNet  MATH  Google Scholar 

  30. H. Whitney, “Analytic extension of differentiable functions defined on closed sets”, Trans. Am. Math. Soc, Vol. 36 (1934), 63–89.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Oldenburg, 26111, Oldenburg, Germany

    Michael Langenbruch

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  1. Michael Langenbruch
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Correspondence to Michael Langenbruch.

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Dedicated to Professor Dr. H. G. Tillmann on the occasion of his 75th birthday

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Langenbruch, M. Analytic Extension of Smooth Functions. Results. Math. 36, 281–296 (1999). https://doi.org/10.1007/BF03322117

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Mathematics Subject Classification: Primary

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