Journal of Applied Mathematics
Volume 2012 (2012), Article ID 806945, 17 pages
Research Article

A Comparison between Fixed-Basis and Variable-Basis Schemes for Function Approximation and Functional Optimization

Department of Communication, Computer and System Sciences (DIST), University of Genova, Via Opera Pia 13, 16145 Genova, Italy

Received 30 July 2011; Accepted 14 November 2011

Academic Editor: Jacek Rokicki

Copyright © 2012 Giorgio Gnecco. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fixed-basis and variable-basis approximation schemes are compared for the problems of function approximation and functional optimization (also known as infinite programming). Classes of problems are investigated for which variable-basis schemes with sigmoidal computational units perform better than fixed-basis ones, in terms of the minimum number of computational units needed to achieve a desired error in function approximation or approximate optimization. Previously known bounds on the accuracy are extended, with better rates, to families of 𝑑 -variable functions whose actual dependence is on a subset of 𝑑 𝑑 variables, where the indices of these 𝑑 variables are not known a priori.