Approximation of fuzzy numbers by convolution method

TL;DR

This paper advances the convolution method to efficiently generate differentiable, Lipschitz, and core-preserving fuzzy number approximations, even with finite non-differentiable points, improving prior techniques.

Contribution

It extends the convolution method to construct smooth, differentiable approximations for a broader class of fuzzy numbers, including those with finite non-differentiable points.

Findings

Constructs differentiable approximations with finite steps.

Provides a method to preserve the core during approximation.

Achieves Lipschitz continuity in the approximations.

Abstract

In this paper we consider how to use the convolution method to construct approximations, which consist of fuzzy numbers sequences with good properties, for a general fuzzy number. It shows that this convolution method can generate differentiable approximations in finite steps for fuzzy numbers which have finite non-differentiable points. In the previous work, this convolution method only can be used to construct differentiable approximations for continuous fuzzy numbers whose possible non-differentiable points are the two endpoints of 1-cut. The constructing of smoothers is a key step in the construction process of approximations. It further points out that, if appropriately choose the smoothers, then one can use the convolution method to provide approximations which are differentiable, Lipschitz and preserve the core at the same time.