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Approximations Involving Lipschitz Functions

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Lipschitz Functions

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2241))

Abstract

In this chapter we study the problem of the uniform approximation of some classes of functions (e.g. uniformly continuous) by Lipschitz functions, based on the existence of Lipschitz partitions of unity or on some extension results for Lipschitz functions. A result due to Baire on the approximation of semi-continuous functions by continuous ones, based on McShane’s extension method, is also included. The chapter ends with a study of homotopy of Lipschitz functions and a brief presentation of Lipschitz manifolds.

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References

  1. D. Azagra, R. Fry, L. Keener, Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces. J. Funct. Anal. 262(1), 124–166 (2012)

    Article  MathSciNet  Google Scholar 

  2. V.I. Bogachev, S.A. Shkarin, Differentiable and Lipschitz mappings of Banach spaces. Mat. Zametki 44(5), 567–583 (1988) [Russian; English translation in Math. Notes 44(5–6), 790–798 (1988)]

    Google Scholar 

  3. M.C. Boiso, Approximation of Lipschitz functions by Δ-convex functions in Banach spaces. Isr. J. Math. 106, 269–284 (1998)

    Article  MathSciNet  Google Scholar 

  4. I. Colojoară, On Whitney’s imbedding theorem. Rev. Roumaine Math. Pures Appl. 10, 291–296 (1965) [Correction in the same journal 10, 1051–1052 (1965)]

    Google Scholar 

  5. I. Colojoară, Lipschitz closed embedding of Hilbert-Lipschitz manifolds. Rend. Mat. Appl. 15(4), 561–568 (1996)

    MathSciNet  MATH  Google Scholar 

  6. R.M. Dudley, Convergence of Baire measures. Studia Math. 27, 251–268, (1966) [Correction: Studia Math. 51, 275 (1974)]

    Article  MathSciNet  Google Scholar 

  7. R. Engelking, General Topology. 2nd edn. Sigma Series in Pure Mathematics, vol. 6 (Heldermann Verlag, Berlin, 1989) [Translated from the Polish by the author]

    Google Scholar 

  8. G. Gagneux, Approximations lipschitziennes de la fonction signe et schémas semi-discrétisés implicites de problèmes quasi-linéaires dégénérés, in Publ. Math. Fac. Sci. Besançon, vol. 8, Besançon (1984), pp. Exp. No. 5 (1984), 16 pp [French]

    Google Scholar 

  9. G. Georganopoulos, Sur l’approximation des fonctions continues par des fonctions lipschitziennes. C. R. Acad. Sci. Paris Sér. A-B 264, A319–A321 (1967)

    MathSciNet  MATH  Google Scholar 

  10. T.G. Honary, H. Mahyar, Approximation in Lipschitz algebras of infinitely differentiable functions. Bull. Korean Math. Soc. 36(4), 629–636 (1999)

    MathSciNet  MATH  Google Scholar 

  11. T.G. Honary, H. Mahyar, Approximation in Lipschitz algebras. Quaest. Math. 23(1), 13–19 (2000)

    Article  MathSciNet  Google Scholar 

  12. S.-T. Hu, On Lipschitz mappings. Port. Math. 7, 45–49 (1948)

    MathSciNet  MATH  Google Scholar 

  13. J. Luukkainen, Extension of spaces, maps, and metrics in Lipschitz topology. Ann. Acad. Sci. Fenn. Math. Diss. No. 17 (1978)

    Google Scholar 

  14. J. Luukkainen, Lipschitz and quasiconformal approximation of homeomorphism pairs. Topol. Appl. 109(1), 1–40 (2001)

    Article  MathSciNet  Google Scholar 

  15. J. Luukkainen, J. Väisälä, Elements of Lipschitz topology. Ann. Acad. Sci. Fenn. Math. 3(1), 85–122 (1977)

    Article  MathSciNet  Google Scholar 

  16. J.H. Michael, Approximation of functions by means of Lipschitz functions. J. Aust. Math. Soc. 3, 134–150 (1963)

    Article  MathSciNet  Google Scholar 

  17. J.H. Michael, Lipschitz approximations to summable functions. Acta Math. 111, 73–95 (1964)

    Article  MathSciNet  Google Scholar 

  18. R. Miculescu, Equivalent definitions for Lipschitz compact connected manifolds. Ann. Univ. Bucureşti Mat. 49(1), 53–62 (2000)

    MathSciNet  MATH  Google Scholar 

  19. R. Miculescu, A LIP immersion of Lipschitz manifolds modelled on some Banach spaces. Bull. Greek Math. Soc. 43, 99–104 (2000)

    MathSciNet  MATH  Google Scholar 

  20. R. Miculescu, Les fonctions lipschitziennes homotopiques sont Lipschitz homotopiques. Rev. Roum. Math. Pures Appl. 45(1), 119–122 (2000)

    MathSciNet  MATH  Google Scholar 

  21. R. Miculescu, Approximation of continuous functions by Lipschitz functions. Real Anal. Exchange 26(1), 449–452 (2000/01)

    MathSciNet  MATH  Google Scholar 

  22. R. Miculescu, Lipschitz approximation of uniformly continuous convex-valued functions. Bull. Greek Math. Soc. 46, 129–132 (2002)

    MathSciNet  MATH  Google Scholar 

  23. R. Miculescu, Approximations by Lipschitz functions generated by extensions. Real Anal. Exchange 28(1), 33–40 (2002/03)

    Article  MathSciNet  Google Scholar 

  24. R. Miculescu, Some observations on generalized Lipschitz functions. Rocky Mountain J. Math. 37(3), 893–903 (2007)

    Article  MathSciNet  Google Scholar 

  25. R. Miculescu, A selection of embedding results for Lipschitz manifolds. Ann. Univ. Buchar. Math. Ser. 1(1), 121–124 (2010)

    MathSciNet  MATH  Google Scholar 

  26. M. Moussaoui, Approximations lipschitziennes de multifonctions. Application aux conditions nécessaires d’optimalité. Sém. Anal. Convexe 20, 34 (1990)

    MathSciNet  MATH  Google Scholar 

  27. C. Mustăţa, On a theorem of Baire about lower semicontinuous functions. Rev. Anal. Numér. Théor. Approx. 37(1), 71–75 (2008)

    MathSciNet  MATH  Google Scholar 

  28. J. Rosenberg, Applications of analysis on Lipschitz manifolds, in Miniconferences on Harmonic Analysis and Operator Algebras (Canberra, 1987), Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 16 (The Australian National University, Canberra, 1988), pp. 269–283

    Google Scholar 

  29. N. Teleman, The index of signature operators on Lipschitz manifolds. Inst. Hautes Études Sci. Publ. Math.(58), 39–78 (1983)

    Article  MathSciNet  Google Scholar 

  30. P. Tukia, Lipschitz approximation of homeomorphisms. Ann. Acad. Sci. Fenn. Ser. A I Math. 4(1), 137–144 (1979)

    Article  MathSciNet  Google Scholar 

  31. P. Tukia, J. Väisälä, Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6(2), 303–342 (1981)

    Article  MathSciNet  Google Scholar 

  32. P. Tukia, J. Väisälä, Bi-Lipschitz extensions of maps having quasiconformal extensions. Math. Ann. 269(4), 561–572 (1984)

    Article  MathSciNet  Google Scholar 

  33. J. Väisälä, Piecewise linear approximation of lipeomorphisms. Ann. Acad. Sci. Fenn. Ser. A I Math. 3(2), 377–383 (1977)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Babeş-Bolyai University, Cluj-Napoca, Romania

    Ştefan Cobzaş & Adriana Nicolae

  2. Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Brasov, Romania

    Radu Miculescu

Authors
  1. Ştefan Cobzaş
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  2. Radu Miculescu
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  3. Adriana Nicolae
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Cobzaş, Ş., Miculescu, R., Nicolae, A. (2019). Approximations Involving Lipschitz Functions. In: Lipschitz Functions. Lecture Notes in Mathematics, vol 2241. Springer, Cham. https://doi.org/10.1007/978-3-030-16489-8_6

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