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1998 | 129 | 3 | 265-284
Tytuł artykułu

On regularization in superreflexive Banach spaces by infimal convolution formulas

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-Hölder derivatives (for some 0 < α≤ 1). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function f which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of Δ-convex $C^{1,α}$ functions converging to f uniformly on bounded sets and preserving the infimum and the set of minimizers of f. The techniques we develop are based on the use of extended inf-convolution formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.
Czasopismo
Rocznik
Tom
129
Numer
3
Strony
265-284
Opis fizyczny
Daty
wydano
1998
otrzymano
1997-06-24
poprawiono
1997-11-03
Twórcy
  • Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160,41080 Sevilla, Spain, boiso@cica.es
  • Equipe d'Analyse, Boîte 186, Tour 46-00, 4-ième étage, Université Pierre et Marie Curie, 4, Pl. Jussieu, 75252 Paris Cedex 05, France
Bibliografia
  • [AA] H. Attouch and D. Azé, Approximation and regularization of arbitrary functions in Hilbert spaces by the Lasry-Lions method, Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 289-312.
  • [AW] H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-729.
  • [BPP] M. Bougeard, J. P. Penot and A. Pommelet, Towards minimal assumptions for the infimal convolution regularization, J. Approx. Theory 64 (1991), 245-270.
  • [C] M. Cepedello-Boiso, Approximation of Lipschitz functions by Δ-convex functions in Banach spaces, Israel J. Math., to appear.
  • [DFH] R. Deville, V. Fonf and P. Hájek, Analytic and $C^k$ approximations of norms in separable Banach spaces, Studia Math. 120 (1996), 61-74.
  • [DGZ] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math. 64, Longman, Boston, 1993.
  • [E] P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Proc. Internat. Sympos. Partial Differential Equations and the Geometry of Normed Linear Spaces II (Jerusalem, 1972), Israel J. Math. 13 (1973), 281-288.
  • [Fa] M. Fabián, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. 51 (1985), 113-126.
  • [Fr] J. Frontisi, Smooth partitions of unity in Banach spaces, Rocky Mountain J. Math. 25 (1995), 1295-1304.
  • [GR] A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Trans. Amer. Math. Soc. 322 (1990), 691-709.
  • [H] J. Hoffman-Jørgensen, On the modulus of smoothness and the $G_*$-conditions in B-spaces, preprint series, Aarhus Universitet, Matematisk Inst., 1974.
  • [L] G. Lancien, On uniformly convex and uniformly Kadec-Klee renormings, Serdica Math. J. 21 (1995), 1-18.
  • [LL] J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986), 257-266.
  • [NS] A. S. Nemirovskiĭ and S. M. Semenov, The polynomial approximation of functions on Hilbert space, Mat. Sb. (N.S.) 92 (1973), 257-281, 344 (in Russian).
  • [Ph] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1993.
  • [Pi] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 236-350.
  • [St_1] T. Strömberg, The operation of infimal convolution, Dissertationes Math. (Rozprawy Mat.) 352 (1996).
  • [St_2] T. Strömberg, On regularization in Banach spaces, Ark. Mat. 34 (1996), 383-406.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv129i3p265bwm
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