American Institute of Mathematical Sciences

Advanced
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

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 & 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.  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), 9-36. 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), 277-295. 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), 289-320. 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), 157-184.  Google Scholar

[6]

C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoam., 21 (2005), 313-348.  Google Scholar

[7]

C. Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam., 23 (2007), 269-280.  Google Scholar

[8]

C. Fefferman, $C^m$ extension by linear operators, Ann. of Math. (2), 166 (2007), 779-835. 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), 1-48.  Google Scholar

[10]

C. Fefferman, Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207-220.  Google Scholar

[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.  Google Scholar

[12]

L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[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.  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), 20 pp.  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), 223-248. doi: 10.1007/PL00001004.  Google Scholar

[16]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.  Google Scholar

[17]

G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.  Google Scholar

[18]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.  Google Scholar

[19]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[20]

M. Shub, "Global Stability of Dynamical Systems,'' Springer-Verlag, New York, 1987.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  Google Scholar

[25]

H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space, Trans. Amer. Math. Soc., 40 (1936), 309-317.  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), 609-622. 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), 61-100. doi: 10.1007/s002200100593.  Google Scholar

[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.  Google Scholar

[29]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.  Google Scholar

show all references

References:
[1]

R. Abraham and J. Robbin, "Transversal Mappings and Flows,'' W.A. Benjamin, Inc., New York-Amsterdam, 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), 9-36. 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), 277-295. 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), 289-320. 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), 157-184.  Google Scholar

[6]

C. Fefferman, Interpolation and extrapolation of smooth functions by linear operators, Rev. Mat. Iberoam., 21 (2005), 313-348.  Google Scholar

[7]

C. Fefferman, The structure of linear extension operators for $C^m$, Rev. Mat. Iberoam., 23 (2007), 269-280.  Google Scholar

[8]

C. Fefferman, $C^m$ extension by linear operators, Ann. of Math. (2), 166 (2007), 779-835. 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), 1-48.  Google Scholar

[10]

C. Fefferman, Whitney's extension problems and interpolation of data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 207-220.  Google Scholar

[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.  Google Scholar

[12]

L. Grafakos, "Classical and Modern Fourier Analysis,'' Pearson Education, Inc., Upper Saddle River, NJ, 2004.  Google Scholar

[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.  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), 20 pp.  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), 223-248. doi: 10.1007/PL00001004.  Google Scholar

[16]

G. Popov, KAM theorem for Gevrey Hamiltonians, Ergodic Theory Dynam. Systems, 24 (2004), 1753-1786. doi: 10.1017/S0143385704000458.  Google Scholar

[17]

G. Popov, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26 (2004), 87-107.  Google Scholar

[18]

J. Pöschel, Integrability of Hamiltonian systems on Cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504.  Google Scholar

[19]

J. Pöschel, On elliptic lower-dimensional tori in Hamiltonian systems, Math. Z., 202 (1989), 559-608. doi: 10.1007/BF01221590.  Google Scholar

[20]

M. Shub, "Global Stability of Dynamical Systems,'' Springer-Verlag, New York, 1987.  Google Scholar

[21]

E. M. Stein, "Singular Integrals and Differentiability Properties of Functions,'' Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, 1970.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[24]

H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), 63-89.  Google Scholar

[25]

H. Whitney, Differentiable functions defined in arbitrary subsets of Euclidean space, Trans. Amer. Math. Soc., 40 (1936), 309-317.  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), 609-622. 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), 61-100. doi: 10.1007/s002200100593.  Google Scholar

[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.  Google Scholar

[29]

E. Zehnder, Generalized implicit function theorems with applications to some small divisor problems. I, Comm. Pure Appl. Math., 28 (1975), 91-140.  Google Scholar

[1]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57
[2]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413
[3]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069
[4]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080
[5]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693
[6]

Luigi Chierchia, Gabriella Pinzari. Properly-degenerate KAM theory (following V. I. Arnold). Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 545-578. doi: 10.3934/dcdss.2010.3.545
[7]

Roman Srzednicki. On periodic solutions in the Whitney's inverted pendulum problem. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2127-2141. doi: 10.3934/dcdss.2019137
[8]

Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197
[9]

Yanzhao Cao, Anping Liu, Zhimin Zhang. Special section on differential equations: Theory, application, and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : i-ii. doi: 10.3934/dcdsb.2015.20.5i
[10]

Frank Trujillo. Uniqueness properties of the KAM curve. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5165-5182. doi: 10.3934/dcds.2021072
[11]

Ryan Alvarado, Irina Mitrea, Marius Mitrea. Whitney-type extensions in quasi-metric spaces. Communications on Pure & Applied Analysis, 2013, 12 (1) : 59-88. doi: 10.3934/cpaa.2013.12.59
[12]

Christopher M. Kellett. Classical converse theorems in Lyapunov's second method. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2333-2360. doi: 10.3934/dcdsb.2015.20.2333
[13]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[14]

Peter Giesl. Converse theorem on a global contraction metric for a periodic orbit. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5339-5363. doi: 10.3934/dcds.2019218
[15]

Alessandra Celletti. Some KAM applications to Celestial Mechanics. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 533-544. doi: 10.3934/dcdss.2010.3.533
[16]

Martin Brokate, Pavel Krejčí. Weak differentiability of scalar hysteresis operators. Discrete & Continuous Dynamical Systems, 2015, 35 (6) : 2405-2421. doi: 10.3934/dcds.2015.35.2405
[17]

Misha Guysinsky, Boris Hasselblatt, Victoria Rayskin. Differentiability of the Hartman--Grobman linearization. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 979-984. doi: 10.3934/dcds.2003.9.979
[18]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481
[19]

Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409
[20]

Xinmin Yang, Xiaoqi Yang. A note on mixed type converse duality in multiobjective programming problems. Journal of Industrial & Management Optimization, 2010, 6 (3) : 497-500. doi: 10.3934/jimo.2010.6.497

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (53)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences