On regularization in superreflexive Banach spaces by infimal convolution formulas
Cepedello-Boiso, Manuel
Studia Mathematica, Tome 129 (1998), p. 265-284 / Harvested from The Polish Digital Mathematics Library
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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 C1,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.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216504 
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     year = {1998},
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Cepedello-Boiso, Manuel. On regularization in superreflexive Banach spaces by infimal convolution formulas. Studia Mathematica, Tome 129 (1998) pp. 265-284. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv129i3p265bwm/

[00000] [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. | Zbl 0780.41021

[00001] [AW] H. Attouch and R. Wets, Quantitative stability of variational systems: I. The epigraphical distance, Trans. Amer. Math. Soc. 328 (1991), 695-729. | Zbl 0753.49007

[00002] [BPP] M. Bougeard, J. P. Penot and A. Pommelet, Towards minimal assumptions for the infimal convolution regularization, J. Approx. Theory 64 (1991), 245-270. | Zbl 0759.49003

[00003] [C] M. Cepedello-Boiso, Approximation of Lipschitz functions by Δ-convex functions in Banach spaces, Israel J. Math., to appear. | Zbl 0920.46010

[00004] [DFH] R. Deville, V. Fonf and P. Hájek, Analytic and Ck approximations of norms in separable Banach spaces, Studia Math. 120 (1996), 61-74. | Zbl 0857.46008

[00005] [DGZ] R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces, Pitman Monographs Surveys Pure Appl. Math. 64, Longman, Boston, 1993.

[00006] [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.

[00007] [Fa] M. Fabián, Lipschitz smooth points of convex functions and isomorphic characterizations of Hilbert spaces, Proc. London Math. Soc. 51 (1985), 113-126. | Zbl 0549.46025

[00008] [Fr] J. Frontisi, Smooth partitions of unity in Banach spaces, Rocky Mountain J. Math. 25 (1995), 1295-1304. | Zbl 0853.46015

[00009] [GR] A. Griewank and P. J. Rabier, On the smoothness of convex envelopes, Trans. Amer. Math. Soc. 322 (1990), 691-709. | Zbl 0712.49010

[00010] [H] J. Hoffman-Jørgensen, On the modulus of smoothness and the G*-conditions in B-spaces, preprint series, Aarhus Universitet, Matematisk Inst., 1974.

[00011] [L] G. Lancien, On uniformly convex and uniformly Kadec-Klee renormings, Serdica Math. J. 21 (1995), 1-18. | Zbl 0837.46011

[00012] [LL] J. M. Lasry and P. L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (1986), 257-266. | Zbl 0631.49018

[00013] [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). | Zbl 0286.41025

[00014] [Ph] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1993. | Zbl 0921.46039

[00015] [Pi] G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), 236-350. | Zbl 0344.46030

[00016] [St_1] T. Strömberg, The operation of infimal convolution, Dissertationes Math. (Rozprawy Mat.) 352 (1996). | Zbl 0858.49010

[00017] [St_2] T. Strömberg, On regularization in Banach spaces, Ark. Mat. 34 (1996), 383-406. | Zbl 0873.49028