Fuzzy Mathematics: Approximation Theory

Frontmatter

INTRODUCTION

Basics

The concept of fuzziness was first discovered and introduced in the seminal article written by Lotfi A. Zadeh in 1965, see [103].

So in our description next we follow [103].

Frequently classes of objects encountered in the real natural world do not have exactly defined criteria of membership. For example, the class of animals clearly includes lions, tigers, horses, birds, fish, etc. as its members and obviously excludes objects such as trees, gases, cars, stones, houses, metals, etc. However there are objects such as starfish, bacteria, etc. that have an ambiguous status in comparison to the class of animals.

Similar ambiguity arises when we compare the number 20 to the class of real numbers much greater than zero. Clearly, “the class of real numbers much greater than zero”, or “the class of beautiful women”, or “the class of tall men” or “the class of smart students” are not defined precisely, thus they do not constitute sets of objects in the usual mathematical sense where each element of a set is 100% there. However such imprecisely considered “classes” of objects exist frequently and play an important role in every aspect of our lives,they show up a lot especially in engineering, computer science, pattern recognition, industry, etc. So the concept under consideration is the fuzzy set, which is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function which assigns to each object a grade of membership varying from zero to one. The notions of inclusion, union, intersection, complementation.

George A. Anastassiou

ABOUT H-FUZZY DIFFERENTIATION

Abstract

The concept of H-fuzzy differentiation is discussed thoroughly in the univariate and multivariate cases. Basic H-derivatives are calculated and then important theorems are presented on the topic, such as, the H-mean value theorem, the univariate and multivariate H-chain rules, and the interchange of the order of H-fuzzy differentiation. Finally is given a multivariate H-fuzzy Taylor formula. This treatment relies in [10].

George A. Anastassiou

ON FUZZY TAYLOR FORMULAE

Abstract

We present Fuzzy Taylor formulae with integral remainder in the univariate and multivariate cases, analogs of the real setting. This chapter is based on [19].

George A. Anastassiou

FUZZY OSTROWSKI INEQUALITIES

Abstract

We present optimal upper bounds for the deviation of a fuzzy continuous function from its fuzzy average over \([a, b] \subset {\mathbb R}\), error is measured in the D-fuzzy metric. The established fuzzy Ostrowski type inequalities are sharp, in fact attained by simple fuzzy real number valued functions. These in- equalities are given for fuzzy Hölder and fuzzy differentiable functions and these facts are reflected in their right-hand sides. This chapter relies on [13].

George A. Anastassiou

A FUZZY TRIGONOMETRIC APPROXIMATION THEOREM OF WEIERSTRASS-TYPE

Abstract

In this chapter we show that any 2π-periodic fuzzy continuous function from \(\mathbb R\) to the fuzzy number space \({\mathbb R}_{\mathcal F}\), can be uniformly approximated by some fuzzy trigonometric polynomials. This chapter is based on [31].

George A. Anastassiou

ON BEST APPROXIMATION AND JACKSON-TYPE ESTIMATES BY GENERALIZED FUZZY POLYNOMIALS

Abstract

In [31] was proved that any 2π-periodic continuous fuzzy-number-valued function can be uniformly approximated by sequences of generalized fuzzy trigonometric polynomials, but without giving any estimates for the approximation error. In this chapter, connected to the best approximation problem we present Jackson-type estimates. For the algebraic case we also give a Jackson-type estimate, using the Szabados-type polynomials. Finally, as an application we study the convergence of fuzzy Lagrange interpolation polynomials. This chapter relies on [41].

George A. Anastassiou

BASIC FUZZY KOROVKIN THEORY

Abstract

We present the basic fuzzy Korovkin theorem via a fuzzy Shisha–Mond inequality given here. This determines the degree of convergence with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The surprising fact is that only the real case Korovkin assumptions are enough for the validity of the fuzzy Korovkin theorem, along with a natural realization condition fulfilled by the sequence of fuzzy positive linear operators. The last condition is fulfilled by almost all operators defined via fuzzy summation or fuzzy integration. This chapter relies on [18].

George A. Anastassiou

FUZZY TRIGONOMETRIC KOROVKIN THEORY

Abstract

We present the fuzzy Korovkin trigonometric theorem via a fuzzy Shisha–Mond trigonometric inequality presented here too. This determines the degree of approximation with rates of a sequence of fuzzy positive linear operators to the fuzzy unit operator. The astonishing fact is that only the real case trigonometric assumptions are enough for the validity of the fuzzy trigonometric Korovkin theorem, along with a very natural realization condition fulfilled by the sequence of fuzzy positive linear operators. The latter condition is satisfied by almost all operators defined via fuzzy summation or fuzzy integration. This chapter is based on [32].

George A. Anastassiou

FUZZY GLOBAL SMOOTHNESS PRESERVATION

Abstract

Here we present the property of global smoothness preservation for fuzzy linear operators acting on spaces of fuzzy continuous functions. Basically we transfer the property of real global smoothness preservation into the fuzzy setting, via some natural realization condition fulfilled by almost all example-fuzzy linear operators. The derived inequalities involve fuzzy moduli of continuity and we give examples. This chapter relies on [21].

George A. Anastassiou

FUZZY KOROVKIN THEORY AND INEQUALITIES

Abstract

Here we study the fuzzy positive linear operators acting on fuzzy continuous functions. We prove the fuzzy Riesz representation theorem, the fuzzy Shisha–Mond type inequalities and fuzzy Korovkin type theorems regarding the fuzzy convergence of fuzzy positive linear operators to the fuzzy unit in various cases. Special attention is paid to the study of fuzzy weak convergence of finite positive measures to the unit Dirac measure. All convergences are with rates and are given via fuzzy inequalities involving the fuzzy modulus of continuity of the engaged fuzzy valued function. The assumptions for the Korovkin theorems are minimal and of natural realization, fulfilled by almost all example – fuzzy positive linear operators. The surprising fact is that the real Korovkin test functions assumptions carry over here in the fuzzy setting and they are the only enough to impose the conclusions of fuzzy Korovkin theorems. We give a lot of examples and applications to our theory, namely: to fuzzy Bernstein operators, to fuzzy Shepard operators, to fuzzy Szasz–Mirakjan and fuzzy Baskakov-type operators and to fuzzy convolution type operators.

We work in general, basically over real normed vector space domains that are compact and convex or just convex. On the way to prove the main theorems we establish a lot of other interesting and important side results This chapter relies on [24].

George A. Anastassiou

HIGHER ORDER FUZZY KOROVKIN THEORY USING INEQUALITIES

Abstract

Here is studied with rates the fuzzy uniform and L _p , p ≥ 1, convergence of a sequence of fuzzy positive linear operators to the fuzzy unit operator acting on spaces of fuzzy differentiable functions. This is done quantitatively via fuzzy Korovkin type inequalities involving the fuzzy modulus of continuity of a fuzzy derivative of the engaged function. From there we deduce general fuzzy Korovkin type theorems with high rate of convergence. The surprising fact is that basic real positive linear operator simple assumptions enforce here the fuzzy convergences. At the end we give applications. The results are univariate and multivariate. The assumptions are minimal and natural fulfilled by almost all example—fuzzy positive linear operators. This chapter follows [20].

George A. Anastassiou

FUZZY WAVELET LIKE OPERATORS

Abstract

The basic wavelet type operators \(A_k, B_k, C_k, D_k, k \in {\mathbb Z}\) were studied extensively in the real case, e.g., see [9]. Here they are extended to the fuzzy setting and are defined similarly via a real valued scaling function. Their pointwise and uniform convergence with rates to the fuzzy unit operator I is presented. The produced Jackson type inequalities involve the fuzzy first modulus of continuity and usually are proved to be sharp, in fact attained. Furthermore all fuzzy wavelet like operators A _k , B _k , C _k , D _k preserve monotonicity in the fuzzy sense. Here we do not suppose any kind of orthogonality condition on the scaling function φ, and the operators act on fuzzy valued continuous functions over \(\mathbb R\). This chapter follows [14].

George A. Anastassiou

ESTIMATES TO DISTANCES BETWEEN FUZZY WAVELET LIKE OPERATORS

Abstract

The basic fuzzy wavelet like operators \(A_k, B_k, C_k, D_k, k \in {\mathbb Z}\) were first introduced in [14], see also Chapter 12, where they were studied among others for their pointwise/uniform convergence with rates to the fuzzy unit operator I. Here we continue this study by estimating the fuzzy distances between these operators. We give the pointwise convergence with rates of these distances to zero. The related approximation is of higher order since we involve these higher order fuzzy derivatives of the engaged fuzzy continuous function f. The derived Jackson type inequalities involve the fuzzy (first) modulus of continuity. Some comparison inequalities are also given so we get better upper bounds to the distances we study. The defining of these operators scaling function ϕ is of compact support in [–a, a], a > 0 and is not assumed to be orthogonal. This chapter is based on [23].

George A. Anastassiou

FUZZY APPROXIMATION BY FUZZY CONVOLUTION OPERATORS

Abstract

Here we study four sequences of naturally arising fuzzy integral operators of convolution type that are integral analogs of known fuzzy wavelet like operators, defined via a scaling function. Their fuzzy convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity. Also their high order fuzzy approximation is given similarly by involving the fuzzy modulus of continuity of the Nth order (N ≥ 1) H-fuzzy derivative of the engaged fuzzy number valued function. The fuzzy global smoothness preservation property of these operators is presented too. This chapter relies on [15].

George A. Anastassiou

DEGREE OF APPROXIMATION OF FUZZY NEURAL NETWORK OPERATORS, UNIVARIATE CASE

Abstract

In this chapter we study the rate of convergence to the unit operator of very specific well described univariate Fuzzy neural network operators of Cardaliaguet–Euvrard and “Squashing” types. These Fuzzy operators arise in a very natural and common way among Fuzzy neural networks. The rates are given through Jackson type inequalities involving the Fuzzy modulus of continuity of the engaged Fuzzy valued function or its derivative in the Fuzzy sense. Also several interesting results in Fuzzy real analysis are presented to be used in the proofs of the main results. This chapter is based on [11].

George A. Anastassiou

HIGHER DEGREE OF FUZZY APPROXIMATION BY FUZZY WAVELET TYPE AND NEURAL NETWORK OPERATORS

Abstract

In this chapter are studied in terms of fuzzy high approximation to the unit several basic sequences of fuzzy wavelet type operators and fuzzy neural network operators. These operators are fuzzy analogs of earlier studied real ones. The produced results generalize earlier real ones into the fuzzy setting. Here the high order fuzzy pointwise convergence with rates to the fuzzy unit operator is established through fuzzy inequalities involving the fuzzy modulus of continuity of the Nth order (N ≥ 1) H-fuzzy derivative of the engaged fuzzy number valued function. At the end we present a related L _p result for fuzzy neural network operators. This chapter is based on [16].

George A. Anastassiou

FUZZY RANDOM KOROVKIN THEOREMS AND INEQUALITIES

Abstract

Here we study the fuzzy random positive linear operators acting on fuzzy random continuous functions. We establish a series of fuzzy random Shisha–Mond type inequalities of L ^q -type 1 ≤ q < ∞ and related fuzzy random Korovkin type theorems, regarding the fuzzy random q-mean convergence of fuzzy random positive linear operators to the fuzzy random unit operator for various cases. All convergences are with rates and are given using the above fuzzy random inequalities involving the fuzzy random modulus of continuity of the engaged fuzzy random function. The assumptions for the Korovkin theorems are minimal and of natural realization, fulfilled by almost all example fuzzy random positive linear operators. The astonishing fact is that the real Korovkin test functions assumptions are enough for the conclusions of the fuzzy random Korovkin theory. We give at the end applications. This chapter follows [22].

George A. Anastassiou

FUZZY-RANDOM NEURAL NETWORK APPROXIMATION OPERATORS, UNIVARIATE CASE

Abstract

In this chapter we study the rate of pointwise convergence in the q-mean to the Fuzzy-Random unit operator of very precise univariate Fuzzy-Random neural network operators of Cardaliaguet. Euvrard and “Squashing” types. These Fuzzy-Random operators arise in a natural and common way among Fuzzy-Random neural networks. These rates are given through Probabilistic-Jackson type inequalities involving the Fuzzy-Random modulus of continuity of the engaged Fuzzy-Random function or its Fuzzy derivatives. Also several interesting results in Fuzzy-Random Analysis are given of independent merit, which are used then in the proofs of the main results of the chapter. This chapter follows [17].

George A. Anastassiou

$\mathcal A$ -SUMMABILITY AND FUZZY KOROVKIN APPROXIMATION

Abstract

The aim of this chapter is to present a fuzzy Korovkin-type approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators. This chapter is based on [27].

George A. Anastassiou

$\mathcal A$ -SUMMABILITY AND FUZZY TRIGONOMETRIC KOROVKIN APPROXIMATION

Abstract

The aim of this chapter is to present a fuzzy trigonometric Korovkin-type approximation theorem by using a matrix summability method. We also study the rates of convergence of fuzzy positive linear operators in trigonometric environment. This chapter is based on [28].

George A. Anastassiou

UNIFORM REAL AND FUZZY ESTIMATES FOR DISTANCES BETWEEN WAVELET TYPE OPERATORS AT REAL AND FUZZY ENVIRONMENT

Abstract

The basic fuzzy wavelet type operators \(A_k; B_k; C_k; D_k; k \in {\mathbb Z}\) were studied in [14], [16], see also Chapters 12, 16, for their pointwise and uniform convergence with rates to the fuzzy unit operator. Also they were studied in [23], see also Chapter 13, in terms of estimating their fuzzy differences and giving their pointwise convergence with rates to zero.

George A. Anastassiou

Backmatter