Document Type
Lecture
Publication Date
3-9-2005
Abstract
We first discuss the necessity of studying fuzziness in the real world and mathematics, as fuzziness stands as a separate genuine feature of our physical and ideal world apart from deterministic and random systems. We introduce the fuzzy real numbers with their properties and the fuzzy functions, and we talk about their calculus and its connections to ordinary analysis. We then present a complete study for the approximation of fuzzy functions by fuzzy wavelet-type operators and fuzzy convolution-type operators. The convergence with respect to the fuzzy metric is given with rates involving the fuzzy modulus of continuity of the associated fuzzy function. Separately, we discuss the properties of the fuzzy modulus of continuity. These approximations derive via fuzzy Jackson-type inequalities that are often proven to be sharp. The approximating operators inherit shape properties of the approximated functions. They can preserve fuzzy monotonicity and fuzzy convexity, and most importantly, they possess the property of preserving fuzzy global smoothness. This last property means that as the operators approximate the function, they cannot oscillate more than the function itself, which is expressed via sharp inequalities. So approximation takes place nicely and tightly.
Relational Format
presentation
Recommended Citation
Anastassiou, George, "Contemporary Fuzzy Approximation Methods" (2005). Analysis Seminar. 52.
https://egrove.olemiss.edu/math_analysis/52
