A Modern Introduction to Fuzzy Mathematics

AFuzzy Approximation

Classical approximation theory of real‐valued continuous functions by algebraic or trigonometric polynomials has been a subject of research for more than two centuries (see, e.g. [72, 164, 194, 286] for a rigorous as well as instructive presentation). Here, we are going to give a brief account of some basic results of approximation theory put in a fuzzy setting.

A.1 Weierstrass and Stone–Weierstrass Approximation Theorems

Let us recall two of the most fundamental questions in classical approximation theory:

1. Can every continuous function, , , be arbitrarily well approximated by algebraic polynomials?
2. Can every continuous periodic function with period be arbitrarily well approximated by trigonometric polynomials?

Both questions, that are actually intrinsically interrelated, were answered affirmatively by Karl Theodor Wilhelm Weierstrass in 1885 through perhaps the most significant result in approximation theory known as Weierstrass approximation theorem which, in its basic form, reads:

Theorem A.1.1  Let a continuous real‐valued function be defined on an ...