Jiménez Sevilla, María del Mar (2008) A note on the range of the derivatives of analytic approximations of uniformly continuous functions on co. Journal of Mathematical Analysis and Applications, 348 (2). pp. 573580. ISSN 0022247X

PDF
229kB 
Official URL: http://www.sciencedirect.com/science/journal/0022247X
Abstract
This paper is a contribution to the body of results concerning the size of the set of derivatives of differentiable functions on a Banach space. The results so far have consisted of examples of highly differentiable bump functions (or functions approximating a given continuous function) whose set of derivatives either is surprisingly small or has a given shape. The paper under review treats the case of analytic smoothness. The main result states that every uniformly continuous function on c0 (more generally on a space with property (K)) can be approximated by a realanalytic function whose set of derivatives is contained in T
p>0 lp. This is a significant step forward, as analytic functions are substantially harder to deal with than C1 smooth ones. Indeed, a local perturbation of an analytic function necessarily changes the values of the function everywhere. Also of particular value is the quite precise and elegant description of the set of derivatives of the
approximating function.
Item Type:  Article 

Uncontrolled Keywords:  Approximation by analytic functions; Range of the derivatives 
Subjects:  Sciences > Mathematics > Functional analysis and Operator theory 
ID Code:  13813 
References:  [1] D. Azagra and M. Cepedello, Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds, Duke Math. J. 124 (2004) 207226. [2] D. Azagra and R. Deville, James' theorem fails for starlike bodies, J. Funct. Anal. 180 (2001), 328346. [3] D. Azagra and M. JiménezSevilla, The failure of Rolle's theorem in in infinitedimensional Banach spaces, J. Funct. Anal. 182 (2001) 207226. [4] D. Azagra and M. JiménezSevilla, Approximation by smooth functions with no critical points on separable Banach spaces, J. Funct. Anal. 242 (2007), no. 1, 136. [5] D. Azagra, R. Deville and M. JiménezSevilla, On the range of the derivatives of a smooth mapping between Banach spaces, Proc. Cambridge Phil. Soc. 134 (2003), no. 1, 163185. [6] J. M. Borwein, M. Fabian and P.D. Loewen, The range of the gradient of a Lipschitz C1smooth bump in infinite dimensions, Isr. J. Math. 132 (2002), 239251. [7] M. Cepedello and P. Hájek Analytic approximations of uniformly continuous functions in real Banach spaces, J. Math. Anal. Appl. 256 (2001), 8098. [8] R. Deville and P. Hájek, On the range of the derivative of Gâteauxsmooth functions on separable Banach spaces, Isr. J. Math. 145 (2005), 257269. [9] M. Fabian, O. F. K. Kalenda and J. Kolár, Filling analytic sets by the derivatives of C1smooth bumps, Proc. Amer. Math. Soc 133 (2005), no. 1, 295303. [10] J. Ferrer, Rolle's Theorem for polynomials of degree four in a Hilbert space, Journal of Mathematical Analysis and Applications 265 (2002), 322331. [11] R. Fry, Analytic approximation on c0, J. Funct. Analysis 158 (1998), no. 2, 509520. [12] T. Gaspari, On the range of the derivative of a realvalued function with bounded support, Studia Math. 153 (2002), no. 1, 8199. [13] P. Hájek, Smooth functions on c0, Israel Journal of Mathematics 104 (1998), 1727. [14] P. Hájek and M. Johanis, Smooth approximations without critical points, Cent. Eur. J. Math. 1 (2003), no. 3, 284291. [15] J. Kurzweil, On approximations in real Banach spaces, Studia Math. 14 (1954), 214231. [16] J. Mújica, Complex analysis in Banach spaces, NorthHolland Mathematical Studies 120, Elsevier (1986). [17] S.A. Shkarin, On Rolle's theorem in infinitedimensional Banach spaces, Translated from: Mat. Zametki 51 (3) (1992) 128136. 
Deposited On:  10 Nov 2011 11:16 
Last Modified:  06 Feb 2014 09:54 
Repository Staff Only: item control page