Abstract
Let φ be a Whitney jet on a closed set F ⊂ ℝ. By Whitney’s extension theorem φ can be extended to an infinitely differentiable function f on ℝ which is real analytic on ℝ F. The main purpose of this article is to show that f can be chosen in such a way that f¦ℝF has a holomorphic continuation to the open set (ℝ F) × iℝ ⊂ ℂ. In the special case that F is a compact interval or a single point we can even achieve that f¦ℝF has a holomorphic continuation to all of \(\hat {\rm C}\setminus F\). In particular, this implies an improvement of the well-known theorem of E. Borel. We also investigate the question when such extensions are given by a so-called extension operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Borel, Sur quelques points de la théorie des fonctions, Ann. Sci. Ecole Norm. Sup. 12 (1895, 9–55.
L. Frerick, Extension operators for spaces of infinitely differentiable Whitney functions, preprint.
L. Frerick and D. Vogt, Analytic extension of differentiable functions defined in closed sets by means of continuous linear operators, Proc. Amer. Math. Soc. 130 (2002), 1775–1777.
M. Langenbruch, Analytic extension of smooth functions, Result. Math. 36 (1999), 281–296.
B. Malgrange, Ideals of Differentiable Functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Oxford University Press, London, 1967.
R. Meise and D. Vogt, Introduction to Functional Analysis, The Clarendon Press, Oxford University Press, New York, 1997.
B. S. Mitjagin, Approximate dimension and bases in nuclear spaces, Uspehi Mat. Nauk 16 (1961, 63–132.
R. Narasimhan, Analysis on Real and Complex Manifolds, North-Holland Publishing Co., Amsterdam, 1985.
G. Pólya and G. Szegö, Problems and Theorems in Analysis. I, Springer-Verlag, Berlin, New York, 1978.
J. F. Ritt, On the derivatives of a function at a point, Ann. of Math. (2) 18 (1916), 18–23.
J. Schmets and M. Valdivia, On the existence of holomorphic functions having prescribed asymptotic expansions, Z. Anal. Anwendungen 13 (1994), 307–327.
J. Schmets and M. Valdivia, On the existence of continuous linear analytic extension maps for Whitney jets, Bull. Polish Acad. Sci. Math. 45 (1997), 359–367.
J. Schmets and M. Valdivia, Holomorphic extension maps for spaces of Whitney jets, preprint.
R. T. Seeley, Extension of C∞ functions defined in a half space, Proc. Amer. Math. Soc. 15 (1964), 625–626.
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc. 36 (1934), 63–89.
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to Professor Wolfgang Luh (Trier) on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Brück, R., Frerick, L. Holomorphic Extensions of Whitney Jets. Results. Math. 43, 56–73 (2003). https://doi.org/10.1007/BF03322721
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF03322721