Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''

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December 2010, 3(4): 623-641. doi: 10.3934/dcdss.2010.3.623

Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''

Xuemei Li ^1, and Rafael de la Llave ^2,

1.	Department of Mathematics, Hunan Normal University, Changsha 410081, China
2.	School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA 30332-0160, United States

Received March 2009 Revised June 2010 Published August 2010

Abstract
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In KAM theory and other areas of analysis, one is often led to consider sums of functions defined in decreasing domains. A question of interest is whether the limit function is differentiable or not.
We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.
We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.

Keywords: converse approximation., KAM theory, Whitney differentiability.

Mathematics Subject Classification: Primary: 46E35, 58C25; Secondary: 41A05, 37J4.

Citation: Xuemei Li, Rafael de la Llave. Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''. Discrete and Continuous Dynamical Systems - S, 2010, 3 (4) : 623-641. doi: 10.3934/dcdss.2010.3.623

References:

[1]	R. Abraham and J. Robbin, "Transversal Mappings and Flows,'' W.A. Benjamin, Inc., New York-Amsterdam, 1967.
[2]	V. I. Arnol$'$d, Proof of a theorem by A.N. Kolmogorov on the persistence of conditionally periodic motions under a small perturbation of the Hamiltonian, Russian Math. Surveys, 18 (1963), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.
[3]	L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems, Nuovo Cimento B (11), 67 (1982), 277-295. doi: 10.1007/BF02721167.
[4]	R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320. doi: 10.1007/BF02096662.
[5]	R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.
[6]	C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoam., 21 (2005), 313-348.
[7]	C. Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam., 23 (2007), 269-280.
[8]	C. Fefferman, $C^m$ extension by linear operators, Ann. of Math. (2), 166 (2007), 779-835. doi: 10.4007/annals.2007.166.779.
[9]	C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators, Rev. Mat. Iberoam., 25 (2009), 1-48.
[10]	C. Fefferman, Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207-220.
[11]	G. Gallavotti, Perturbation theory for classical Hamiltonian systems, in "Scaling and Self-Similarity in Physics (Bures-sur-Yvette, 1981/1982)'' (ed. J. Fröhlich), Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, 359-426.
[12]	L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson Education, Inc., Upper Saddle River, NJ, 2004.
[13]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995.
[14]	M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets, Math. Phys. Electron. J., 13 (2007), 20 pp.
[15]	G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 223-248. doi: 10.1007/PL00001004.
[16]	G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.
[17]	G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.
[18]	J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.
[19]	J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.
[20]	M. Shub, "Global Stability of Dynamical Systems,'' Springer-Verlag, New York, 1987.
[21]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.
[22]	J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'' Ph.D thesis, University of Texas at Austin, 2002, available at mp_arc # 02-276.
[23]	F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dyn. Syst., 18 (2003), 159-163. doi: 10.1080/1468936031000117857.
[24]	H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.
[25]	H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space, Trans. Amer. Math. Soc., 40 (1936), 309-317.
[26]	J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition, J. Differential Equations, 235 (2007), 609-622. doi: 10.1016/j.jde.2006.12.001.
[27]	X. Yuan, Construction of quasi-periodic breathers via KAM technique, Comm. Math. Phys., 226 (2002), 61-100. doi: 10.1007/s002200100593.
[28]	X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274. doi: 10.1016/j.jde.2005.12.012.
[29]	E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.

show all references

References:

[1]	R. Abraham and J. Robbin, "Transversal Mappings and Flows,'' W.A. Benjamin, Inc., New York-Amsterdam, 1967.
[2]	V. I. Arnol$'$d, Proof of a theorem by A.N. Kolmogorov on the persistence of conditionally periodic motions under a small perturbation of the Hamiltonian, Russian Math. Surveys, 18 (1963), 9-36. doi: 10.1070/RM1963v018n05ABEH004130.
[3]	L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems, Nuovo Cimento B (11), 67 (1982), 277-295. doi: 10.1007/BF02721167.
[4]	R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150 (1992), 289-320. doi: 10.1007/BF02096662.
[5]	R. de la Llave and R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5 (1999), 157-184.
[6]	C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoam., 21 (2005), 313-348.
[7]	C. Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam., 23 (2007), 269-280.
[8]	C. Fefferman, $C^m$ extension by linear operators, Ann. of Math. (2), 166 (2007), 779-835. doi: 10.4007/annals.2007.166.779.
[9]	C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators, Rev. Mat. Iberoam., 25 (2009), 1-48.
[10]	C. Fefferman, Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207-220.
[11]	G. Gallavotti, Perturbation theory for classical Hamiltonian systems, in "Scaling and Self-Similarity in Physics (Bures-sur-Yvette, 1981/1982)'' (ed. J. Fröhlich), Progr. Phys., vol. 7, Birkhäuser Boston, Boston, MA, 1983, 359-426.
[12]	L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson Education, Inc., Upper Saddle River, NJ, 2004.
[13]	A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'' Encyclopedia of Mathematics and its Applications, vol. 54, Cambridge University Press, Cambridge, 1995.
[14]	M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets, Math. Phys. Electron. J., 13 (2007), 20 pp.
[15]	G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms, Ann. Henri Poincaré, 1 (2000), 223-248. doi: 10.1007/PL00001004.
[16]	G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.
[17]	G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.
[18]	J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.
[19]	J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.
[20]	M. Shub, "Global Stability of Dynamical Systems,'' Springer-Verlag, New York, 1987.
[21]	E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.
[22]	J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'' Ph.D thesis, University of Texas at Austin, 2002, available at mp_arc # 02-276.
[23]	F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma, Dyn. Syst., 18 (2003), 159-163. doi: 10.1080/1468936031000117857.
[24]	H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.
[25]	H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space, Trans. Amer. Math. Soc., 40 (1936), 309-317.
[26]	J. Xu and J. You, Gevrey-smoothness of invariant tori for analytic nearly integrable Hamiltonian systems under Rüssmann's non-degeneracy condition, J. Differential Equations, 235 (2007), 609-622. doi: 10.1016/j.jde.2006.12.001.
[27]	X. Yuan, Construction of quasi-periodic breathers via KAM technique, Comm. Math. Phys., 226 (2002), 61-100. doi: 10.1007/s002200100593.
[28]	X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations, J. Differential Equations, 230 (2006), 213-274. doi: 10.1016/j.jde.2005.12.012.
[29]	E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.

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