Fuzzy Interpolation with Extensional Fuzzy Numbers

The article develops further directions stemming from the arithmetic of extensional fuzzy numbers. It presents the existing knowledge of the relationship between the arithmetic and the proposed orderings of extensional fuzzy numbers—so-called $\mathcal{S}$-orderings—and investigates distinct properties of such orderings. The desirable investigation of the $\mathcal{S}$-orderings of extensional fuzzy numbers is directly used in the concept of $\mathcal{S}$-function—a natural extension of the notion of a function that, in its arguments as well as results, uses extensional fuzzy numbers. One of the immediate subsequent applications is fuzzy interpolation. The article provides readers with the basic fuzzy interpolation method, investigation of its properties and an illustrative experimental example on real data. The goal of the paper is, however, much deeper than presenting a single fuzzy interpolation method. It determines direction to a wide variety of fuzzy interpolation as well as other analytical methods stemming from the concept of $\mathcal{S}$-function and from the arithmetic of extensional fuzzy numbers in general.