A
Fuzzy 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, images, images, be arbitrarily well approximated by algebraic polynomials?
  2. Can every continuous periodic function with period images 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:

Equivalently, the supremum norm1 can be used, so that Theorem A.1.1 can be formulated in the following equivalent form:

The algebraic polynomials images are dense in images with respect to the supremum norm, and every continuous function images can be arbitrarily well approximated with respect to the supremum norm, that is, there is an algebraic polynomial images such that

(A.1)equation

A constructive proof (by use of Korovkin sequences and Bernstein polynomials) of Theorem A.1.1 can be found, for example, in [72, 165]. Now, Theorem A.1.1 leads to the corollary that the linear space images is separable, i.e. the real polynomials are dense and each one of them can be approximated by a polynomial with rational coefficients. Further, it can be proved that the aforementioned theorem can be “transferred” to trigonometric polynomials as well, whereby the space images is replaced by the linear space of continuous images‐periodic functions and the interval images is replaced by images (see, e.g. [165] for a proof).

The first generalization of the Weierstrass approximation theorem was formulated in 1937 by Marshall Harvey Stone [267], who replaced the interval images by a compact Hausdorff space and the algebra of real polynomials by a more general (sub)algebra. The Stone–Weierstrass theorem, as it is called, states the following:

For a proof see, for example [88, 268].

A.2 Weierstrass and Stone–Weierstrass Fuzzy Analogs

Before we present the fuzzy analogs for the two theorems presented in the previous section, let us introduce some basic concepts and notions. We follow [9, 130], written by perhaps the leading researchers on fuzzy approximation theory. We begin with a possible definition of a fuzzy function that is more appropriate for our purpose, in relation to the fuzzified graph of an ordinary function [129]:

Consequently, if images is an algebraic polynomial images, then the pair images is a fuzzy algebraic polynomial from images to images, and

equation

where now images is a bivariate algebraic polynomial in images and images.

Similarly, if images is a trigonometric polynomial images, then the pair images is a real‐valued fuzzy trigonometric polynomial, and

equation

where images is a trigonometric polynomial with respect to images and an algebraic polynomial with respect to images.

The following two definitions are more familiar and consistent with Definition A.2.1:

We will further need the notion of distance between two fuzzy functions. Recall that we have already given the Hausdorff distance in Definition 10.2.3 and have discussed the distance between two fuzzy points in Section 11.1. The distance images between two fuzzy functions images and images as given in Definition A.2.1, is the Hausdorff distance images between the sets

equation

and

equation

Hence, we have

In fact, if images and images are compact sets, then images is a metric [131].

Now we have everything needed to come to the fuzzy analog of the Weierstrass theorem (of which a non‐constructive proof is given in [129]):

Let us turn to the fuzzy analog of the Stone–Weierstrass theorem. To this purpose, the interval images is replaced by images, where images is a compact metric space (see Definition 10.1.2 for a metric space) and images is a distance. Consider two subalgebras images and images with images, images. Then, the fuzzy analog of the Stone–Weierstrass theorem states the following [130]:

Naturally, if images is replaced by the real interval images, the aforementioned fuzzy analog of the Stone–Weierstrass theorem gives its place to the fuzzy analog of the Weierstrass theorem, Theorem A.2.1.

In this brief presentation, we have left out some further important issues of fuzzy approximation theory such as quantitative estimates of the approximation error, fuzzy interpolation by use of fuzzy Lagrange interpolating polynomials and splines, and fuzzy Taylor expansion, to name but a few still significant topics. For a thorough presentation, the reader is referred to [9, 130].

Note

  1. 1   A supremum norm is defined as images, images, where images, images is a compact set and images is the normed space of continuous functions images. In fact, images equipped with the supremum norm is complete, that is, it is a Banach space.
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