The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

Abstract

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of α-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

Keywords

• fuzzy norm
• <i>α</i>-fuzzy distance
• fuzzy Kolmogorov <i>n</i>-width
• fuzzy linear <i>n</i>-width