### Impacto

Llavona, José G.
(2006)
*Approximation methods by regular functions.*
Mediterranean Journal of Mathematics , 3
(2 ).
259-271 .
ISSN 1660-5446

PDF
Restringido a Repository staff only hasta 31 December 2020. 207kB |

Official URL: http://www.springerlink.com/content/w41221q040408316/fulltext.pdf

## Abstract

This paper is a survey of approximation results and methods by smooth functions in Banach spaces. The topics considered in the paper are the following: approximation by polynomials by C-k-functions using the method of smooth partitions of unity, approximation by the fine topology, analytic approximation and regularization in Banach spaces using the infimal convolution method.

Item Type: | Article |
---|---|

Uncontrolled Keywords: | Approximation; Differentiability; Polynomials; Banach-spaces; Differentiable functions; Manifolds; Algebras |

Subjects: | Sciences > Mathematics > Functional analysis and Operator theory |

ID Code: | 15946 |

References: | R.M. Aron, Polynomial approximation and a question of G.E. Shilov, Approximation theory and functional analysis (Proc. Internat. Sympos. Approximation Theory, Univ. Estadual de Campinas, Campinas, 1977), J.B. Prolla, North-Holland Math.Stud. (35), North-Holland, Amsterdam-New York (1979), 1-12. D. Azagra and M. Cepedello-Boiso, Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds, Duke Math. J. 124 (no. 1) (2004), 47-66. D. Azagra, J. Gómez, J.A. Jaramillo, M. Lovo and R. Fry, C1-fine approximation of functions on Banach spaces with unconditional basis, Quart. J. Math. 56 (no. 1)(2005), 13-20. D. Azagra and M. Jiménez, Approximation by smooth functions with no critical points on separable infinite dimensional Banach spaces, preprint. R.M. Aron and J.G. Llavona, Composition of weakly uniformly continuous functions, Proc. Roy. Irish Acad. Sect A 88 (no. 1) (1988), 29–33. R.M. Aron and J.B. Prolla, Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math. 313 (1980), 195–216. R. Bonic and J. Frampton, Smooth functions on Banach manifolds, J. Math. Mech. 15 (1966), 877–898. M. Cepedello-Boiso, Approximation of Lipschitz Functions by Δ-convex functions in Banach Spaces, Israel J. Math. 106 (1998), 269–284. M. Cepedello-Boiso, On regularization in superreflexive Banach spaces by infimal convolution formulas, Studia Math. 129 (no. 3) (1998), 265–284. M. Cepedello-Boiso and P. Hájek, Analytic approximations of uniformly continuous functions in real Banach spaces ,J. Math. Anal. Appl. 256 (no. 1) (2001), 80–98. R. Deville, Geometrical implications of the existence of very smooth bump functions in Banach spaces, Israel J. Math. 67 (no. 1) (1989), 1–22. R. Deville, V. Fonf and P. Hájek, Analytic and polyhedral approximation of convex bodies in separable polyhedral Banach spaces, Israel J. Math. 105 (1998), 139–154. R. Deville, V. Fonf and P. Hájek, Analytic and Ck approximations of norms in separable Banach spaces, Studia Math. 120 (no. 1) (1996), 61–74. R. Deville, G. Godefroy and V. Zizler, Smoothness and renormings in Banach spaces, Pitman Monogr. and Surveys in Pure Appl. Math. (64), Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc., New York, 1993. J. Eells and J. McAlpin, An approximate Morse-Sard theorem, J. Math. Mech. 17 (1967/1968), 1055–1064. J.M. Gutierrez and J.G. Llavona, Composition operators between algebras of differentiable functions, Trans. Amer. Math. Soc. 338 (no. 2) (1993), 769–782. R. Haydon, A counterexample to several questions about scattered compact spaces, Bull. London Math. Soc. 22 (no. 3) (1990), 261–268. R. Haydon, Trees in renorming theory, Proc. London Math. Soc. (3) 78 (no. 3) (1999), 541–584. J. Kurzweil, On approximation in real Banach spaces, Studia Math. 14 (1954), 213– 231. J. Kurzweil, On approximation in real Banach spaces by analytic operations, Studia Math. 16 (1957), 124–129. J. Lesmes, On the approximation of continuously differentiable functions in Hilbert spaces, Rev. Colombiana Mat. 8 (1974), 217–223. J.G. Llavona and J.A. Jaramillo, Homomorphisms between algebras of continuous functions, Canad. J. Math. 41 (no. 2) (1989), 132–162. J.G. Llavona, Approximation of differentiable functions, Adv. in Math. Suppl. Stud. (4), Academic Press, New York-London (4) (1979), 197–221. J.M. Lasry and P.L. Lions, A remark on regularization in Hilbert spaces, Israel J. Math. 55 (no. 3) (1986), 257–266. N. Moulis, Approximation de fonctions différentiables sur certains espaces de Banach, (in French) Ann. Inst. Fourier (Grenoble) 21 (no. 4) (1971), 293–345. J. Mujica, Complex Analysis in Banach Spaces. Holomorphic functions and domains of holomorphy in finite and infinite dimensions., North-Holland Math. Stud. (120), North Holland, Publishing Co., Amsterdam, 1986. L. Nachbin, Sur les algébres denses de fonctions différentiables sur une variété, (in French), C.R. Acad. Sci. Paris 228 (1949), 1549-1551. A.M. Nemirovskii and S.M. Semenov, On polynomial approximation of functions on Hilbert space, Mat. USSR Sbornik 21, (1973). H. Torunczyk, Smooth partitions of unity on some non-separable Banach spaces, Studia Math. 46 (1973), 43–51. S.L. Trojanski, On locally uniformly convex and differentiable norms in certain nonseparable Banach spaces, Studia Math. 37 (1970/1971), 173-180. J. Wells, Differentiable functions in c0, Bull. Amer. Math. Soc. 75 (1969), 117–118. H.Whitney, On ideals of differentiable functions, Amer. J. Math. 70 (1948), 635–658. D. Wulbert, Approximation by Ck-functions. Approximation theory, Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1973, ed., Acad. Press, New York (1973), 217–239. |

Deposited On: | 13 Jul 2012 07:18 |

Last Modified: | 06 Feb 2014 10:35 |

Repository Staff Only: item control page