Some Fuzzy-Wavelet-Like Operators and Their Convergence

Firstly, we define some new fuzzy-wavelet-like operators via a real-valued scaling function to approximate the continuous fuzzy functions of one and two variables. Then, by using the modulus of continuity, we prove their pointwise and uniform convergence with rates to the fuzzy unit operator I. Using these fuzzy-wavelet-like operators, we present some numerical examples to illustrate the applicability of the proposed method. Also, we give a new method to approximate the integration of continuous fuzzy realnumber-valued function of two variables by using the fuzzy-wavelet-like operator.


Introduction
Approximating functions in a given space is an old problem.For this purpose, many authors have studied approximation of fuzzy functions on a finite set of distinct points; see [1][2][3][4][5][6].For instance, the authors of [2] proposed a new method to approximate fuzzy functions by distance method.Indeed, they considered the problem for fuzzy data and fuzzy functions using the defuzzification function introduced by Fortemps and Roubens.Also, they introduced a fuzzy polynomial approximation as -approximation of a fuzzy function on a discrete set of points.
Wavelet theory is a relatively new and an emerging area in mathematical research.Also, wavelets are the suitable and powerful tool for approximating functions based on wavelet basis functions.In [1], the author defined some fuzzywavelet-like operators and presented their pointwise and uniform convergence with rates to the fuzzy unit operator .Recently, the authors of [7] constructed fuzzy Haar wavelet and applied it on solving linear fuzzy Fredholm integral equation of second kind.
Here, we propose some new fuzzy-wavelet-like operators via a real-valued scaling function to approximate the continuous fuzzy functions of one and two variables.Also, we prove their pointwise and uniform convergence with rates to the fuzzy unit operator  by using modulus of continuity.It is noticed that we do not suppose any kind of orthogonality condition on the scaling function, and the operators act on fuzzy-valued continuous functions over R and R 2 .
The rest of the paper is organized as follows.In Section 2, we review some elementary concepts of the fuzzy set theory and modulus of continuity.In Section 3, we prepare some theorems for pointwise and uniform convergence of defined fuzzy-wavelet-like operators with rates to the fuzzy unit operator .In Section 4, we give two numerical examples for applicability of the proposed methods.In Section 5, we present an application to approximate the integration of continuous fuzzy real-number-valued function of two variables by using the fuzzy-wavelet-like operator.Finally, this paper is concluded in Section 6.
The set of all fuzzy numbers is denoted by R  .
It is well known that the addition and multiplication operations of real numbers can be extended to R  .In other words, for , V ∈ R  , and  ∈ R, one defines uniquely the sum  ⊕ V and the product  ⊙  by where means the usual addition of two intervals (as subsets of R) and []  means the usual product between a scalar and a subset of R. One uses the same symbol ∑ both for the sum of real numbers and for the sum ⊕ (when the terms are fuzzy numbers).
Definition 8 (see [14]).A fuzzy real number valued function  : R → R  is said to be continuous in  0 ∈ R, if for each  > 0 there is  > 0 such that ((), ( 0 )) < , whenever  ∈ R and | −  0 | < .One says that  is fuzzy continuous on R if  is continuous at each  0 ∈ R and denotes the space of all such functions by   (R).
Similarly we have the following.Definition 10.Let  : R 2 → R  .One calls  a uniformly continuous fuzzy real number valued function, if and only if for any  > 0 there exists  > 0 whenever: One denotes it as  ∈    (R 2 ).
Definition 11 (see [14]).Let  : R → R  , then  is called a nondecreasing function if and only if whenever Definition 12 (see [10,15]).Let  : R → R  be a bounded function, then function  R (, ⋅) : where R + is the set of positive real numbers, is called the modulus of continuity of  on R.
For  : R 2 → R  , the modulus of continuity  R 2 (, ) is defined as follows: where  ∈   (R 2 ) and  > 0. Observe that, for all  ∈   ([, ] 2 ) and ,  > 0, one has where [] is defined to be the greatest integer less than or equal to .Some properties of the modulus of continuity are presented below.
Theorem 14.Let  ∈   (R) and the scaling function () a real-valued bounded function with supp which is a fuzzy-wavelet-like operator.Then for all  ∈ R and  ∈ Z.If  ∈    (R), then as  → +∞ one gets  R (, /2  ) → 0 and lim  → +∞    = , pointwise and uniformly with rates.
Proof.We need to estimate Here, we have Now, by using the above inequality in ( * * ), we can write and hence Theorem 17.All assumptions here are as in Theorem 14. Define for  ∈ Z,  ∈ R the fuzzy-wavelet-like operator where When  ∈    (R) then as  → +∞ one gets  R (, ( + 2)/2  ) → 0 and lim  → +∞ 3   = .
Proof.We have the following: ) . ( This completes the proof.
Proof.We have the following: ( According to this fact that This completes the proof. Theorem 20.Let  : R → R  and the scaling function () a real-valued bounded function with supp  ⊆ [−, ], 0 <  < +∞, such that there exists ,  ∈ R such that  is nondecreasing for  ≤  and  is nonincreasing for  ≥  (the above imply  ≥ 0).Let (, ) be nondecreasing fuzzy function, then (3  )() is nondecreasing fuzzy valued functions for any  ∈ Z.
Proof.The proof is similar to the proof of [15,Theorem 12.14].
Next we observe that So whenever  1 ≤  2 we get Therefore, 3   is nondecreasing.
Theorem 21.Let  and  be as in Theorem 20.Then (1  )() is a nondecreasing fuzzy valued function for any  ∈ Z.
Proof.The proof is similar to the proof of Theorem 20.

Numerical Examples
In this section, we present two examples.Also, we apply the following scaling function [1]: .Now, we apply the fuzzy-wavelet-like operator 2   defined in Theorem 18 to approximate (, ).We also compare the result of using 2   with (, ) in  = 0.9,  = 0.9.For more details, see Figures 2 and 3.

An Application
In this section, we approximate the integration of continuous fuzzy real number valued function of two variables by using the fuzzy-wavelet-like operator 2   defined in Theorem 18.Consider the following fuzzy integral: By using the fuzzy-wavelet-like operator 2  , we get So, we have 2  ) . (54) Proof.We have the following: Mathematical Problems in Engineering Example 3. Consider the following fuzzy integral: where  = (1 + , 3 − ).The exact solution of this example is ( 1 (),  2 ()), 0 ≤  ≤ 1, where (57) Let 2   = (2  (), 2  ()), 0 ≤  ≤ 1.By using proposed method in (53), we present approximate solution to this example for different values of .To compare the exact and the approximate solutions, see Tables 1, 2, and 3.

Conclusions
To approximate the continuous fuzzy functions of one and two variables, some new fuzzy-wavelet-like operators via a real-valued scaling function are defined.Convergence of defined fuzzy like operators is investigated by using the  modulus of continuity.To approximate the integration of continuous fuzzy real number valued function of two variables by using the fuzzy-wavelet-like operator 2   defined in Theorem 18, an application is presented.