# American Institute of Mathematical Sciences

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''

 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

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.
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 & 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, (1967).   Google Scholar [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.  doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar [3] L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems,, Nuovo Cimento B (11), 67 (1982), 277.  doi: 10.1007/BF02721167.  Google Scholar [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.  doi: 10.1007/BF02096662.  Google Scholar [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.   Google Scholar [6] C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators,, Rev. Mat. Iberoam., 21 (2005), 313.   Google Scholar [7] C. Fefferman, The structure of linear extension operators for $C^m$,, Rev. Mat. Iberoam., 23 (2007), 269.   Google Scholar [8] C. Fefferman, $C^m$ extension by linear operators,, Ann. of Math. (2), 166 (2007), 779.  doi: 10.4007/annals.2007.166.779.  Google Scholar [9] C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators,, Rev. Mat. Iberoam., 25 (2009), 1.   Google Scholar [10] C. Fefferman, Whitney's extension problems and interpolation of data,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207.   Google Scholar [11] G. Gallavotti, Perturbation theory for classical Hamiltonian systems,, in, 7 (1981), 359.   Google Scholar [12] L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004).   Google Scholar [13] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar [14] M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets,, Math. Phys. Electron. J., 13 (2007).   Google Scholar [15] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223.  doi: 10.1007/PL00001004.  Google Scholar [16] G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory Dynam. Systems, 24 (2004), 1753.  doi: 10.1017/S0143385704000458.  Google Scholar [17] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.   Google Scholar [18] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Comm. Pure Appl. Math., 35 (1982), 653.  doi: 10.1002/cpa.3160350504.  Google Scholar [19] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559.  doi: 10.1007/BF01221590.  Google Scholar [20] M. Shub, "Global Stability of Dynamical Systems,'', Springer-Verlag, (1987).   Google Scholar [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, 30 (1970).   Google Scholar [22] J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'', Ph.D thesis, (2002), 02.   Google Scholar [23] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159.  doi: 10.1080/1468936031000117857.  Google Scholar [24] H. Whitney, Analytic extensions of differentiable functions defined in closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.   Google Scholar [25] H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space,, Trans. Amer. Math. Soc., 40 (1936), 309.   Google Scholar [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.  doi: 10.1016/j.jde.2006.12.001.  Google Scholar [27] X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.  doi: 10.1007/s002200100593.  Google Scholar [28] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 230 (2006), 213.  doi: 10.1016/j.jde.2005.12.012.  Google Scholar [29] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91.   Google Scholar

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##### References:
 [1] R. Abraham and J. Robbin, "Transversal Mappings and Flows,'', W.A. Benjamin, (1967).   Google Scholar [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.  doi: 10.1070/RM1963v018n05ABEH004130.  Google Scholar [3] L. Chierchia and G. Gallavotti, Smooth prime integrals for quasi-integrable Hamiltonian systems,, Nuovo Cimento B (11), 67 (1982), 277.  doi: 10.1007/BF02721167.  Google Scholar [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.  doi: 10.1007/BF02096662.  Google Scholar [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.   Google Scholar [6] C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators,, Rev. Mat. Iberoam., 21 (2005), 313.   Google Scholar [7] C. Fefferman, The structure of linear extension operators for $C^m$,, Rev. Mat. Iberoam., 23 (2007), 269.   Google Scholar [8] C. Fefferman, $C^m$ extension by linear operators,, Ann. of Math. (2), 166 (2007), 779.  doi: 10.4007/annals.2007.166.779.  Google Scholar [9] C. Fefferman, Extension of $C^{m,\omega}$-smooth functions by linear operators,, Rev. Mat. Iberoam., 25 (2009), 1.   Google Scholar [10] C. Fefferman, Whitney's extension problems and interpolation of data,, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207.   Google Scholar [11] G. Gallavotti, Perturbation theory for classical Hamiltonian systems,, in, 7 (1981), 359.   Google Scholar [12] L. Grafakos, "Classical and Modern Fourier Analysis,'', Pearson Education, (2004).   Google Scholar [13] A. Katok and B. Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems,'', Encyclopedia of Mathematics and its Applications, 54 (1995).   Google Scholar [14] M. Nicol and A. Török, Whitney regularity for solutions to the coboundary equation on Cantor sets,, Math. Phys. Electron. J., 13 (2007).   Google Scholar [15] G. Popov, Invariant tori, effective stability, and quasimodes with exponentially small error terms. I. Birkhoff normal forms,, Ann. Henri Poincaré, 1 (2000), 223.  doi: 10.1007/PL00001004.  Google Scholar [16] G. Popov, KAM theorem for Gevrey Hamiltonians,, Ergodic Theory Dynam. Systems, 24 (2004), 1753.  doi: 10.1017/S0143385704000458.  Google Scholar [17] G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians,, Mat. Contemp., 26 (2004), 87.   Google Scholar [18] J. Pöschel, Integrability of Hamiltonian systems on Cantor sets,, Comm. Pure Appl. Math., 35 (1982), 653.  doi: 10.1002/cpa.3160350504.  Google Scholar [19] J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems,, Math. Z., 202 (1989), 559.  doi: 10.1007/BF01221590.  Google Scholar [20] M. Shub, "Global Stability of Dynamical Systems,'', Springer-Verlag, (1987).   Google Scholar [21] E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'', Princeton Mathematical Series, 30 (1970).   Google Scholar [22] J. A. Vano, "A Nash-Moser Implicit Function Theorem with Whitney Regularity and Applications,'', Ph.D thesis, (2002), 02.   Google Scholar [23] F. Wagener, A note on Gevrey regular KAM theory and the inverse approximation lemma,, Dyn. Syst., 18 (2003), 159.  doi: 10.1080/1468936031000117857.  Google Scholar [24] H. Whitney, Analytic extensions of differentiable functions defined in closed sets,, Trans. Amer. Math. Soc., 36 (1934), 63.   Google Scholar [25] H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space,, Trans. Amer. Math. Soc., 40 (1936), 309.   Google Scholar [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.  doi: 10.1016/j.jde.2006.12.001.  Google Scholar [27] X. Yuan, Construction of quasi-periodic breathers via KAM technique,, Comm. Math. Phys., 226 (2002), 61.  doi: 10.1007/s002200100593.  Google Scholar [28] X. Yuan, Quasi-periodic solutions of completely resonant nonlinear wave equations,, J. Differential Equations, 230 (2006), 213.  doi: 10.1016/j.jde.2005.12.012.  Google Scholar [29] E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I,, Comm. Pure Appl. Math., 28 (1975), 91.   Google Scholar
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