Some Fuzzy-Wavelet-Like Operators and Their Convergence

Abstract

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 real-number-valued function of two variables by using the fuzzy-wavelet-like operator.

1. 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 [1–6]. 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 𝐷-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 fuzzy-wavelet-like operators and presented their pointwise and uniform convergence with rates to the fuzzy unit operator 𝐼. 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 𝐼 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 ℝ².

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 𝐼. 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.

2. Preliminaries

Definition 1 (see [8]). A fuzzy number is a function 𝑢∶ℝ→[0,1] with the following properties: (1)𝑢is normal; that is, there exists𝑥₀∈ℝ such that 𝑢(𝑥₀)=1,(2)𝑢 is fuzzy convex set ()𝑦𝑢(𝑥)𝑦[],i.e.,𝑢𝜆𝑥+1−𝜆≥min,𝑢∀𝑥,𝑦∈ℝ,𝜆∈0,1 (1)(3)𝑢 is upper semicontinuous on ℝ, (4)the {𝑥∈ℝ∶𝑢(𝑥)>0} is compact set.

The set of all fuzzy numbers is denoted by ℝ_𝐹.

Definition 2 (see [9]). Suppose that 𝑢∈ℝ_𝐹. The 𝑟-level set of 𝑢 is denoted by [𝑢]^𝑟=[𝑢₋^(𝑟),𝑢₊^(𝑟)] and defined by [𝑢]^𝑟={𝑥∈ℝ;𝑢(𝑥)≥𝑟}, where 0<𝑟≤1. Also, [𝑢]⁰ is called the support of 𝑢, denoted by supp𝑢, and it is given as [𝑢]⁰={𝑥∈ℝ;𝑢(𝑥)>0}. It follows that the level sets of 𝑢 are closed and bounded intervals in ℝ.
It is well known that the addition and multiplication operations of real numbers can be extended to ℝ_𝐹. In other words, for 𝑢,𝑣∈ℝ_𝐹, and 𝜆∈ℝ, one defines uniquely the sum 𝑢⊕𝑣 and the product 𝜆⊙𝑢 by [𝑢⊕𝑣]^𝑟=[𝑢]^𝑟+[𝑣]^𝑟,[𝜆⊙𝑢]^𝑟=𝜆[𝑢]^𝑟,∀𝑟∈[0,1], (2) where [𝑢]^𝑟+[𝑣]^𝑟 means the usual addition of two intervals (as subsets of ℝ) and 𝜆[𝑢]^𝑟 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 3 (see [9]). An arbitrary fuzzy number is represented, in parametric form, by an ordered pair of functions (𝑢(𝑟),𝑢(𝑟)), 0≤𝑟≤1, which satisfy the following requirements: (1)𝑢(𝑟) is a bounded left continuous nondecreasing function over [0,1], (2)𝑢(𝑟) is a bounded left continuous nonincreasing function over [0,1], (3)𝑢(𝑟)≤𝑢(𝑟), 0≤𝑟≤1.

The addition and scaler multiplication of fuzzy numbers in ℝ_𝐹 are defined as follows: (1) 𝑢⊕𝑣=(𝑢(𝑟)+𝑣(𝑟),𝑢(𝑟)+𝑣(𝑟)),(2)(𝜆⊙𝑢)=𝜆𝑢(𝑟),𝜆𝑢(𝑟)𝜆𝜆≥0,𝑢(𝑟),𝜆𝑢(𝑟)𝜆<0. (3)

Definition 4 (see [10]). For arbitrary fuzzy numbers 𝑢, 𝑣, the quantity 𝐷(𝑢,𝑣)=sup_[]𝑟∈0,1max𝑢₋^(𝑟)−𝑣₋^(𝑟),𝑢₊^(𝑟)−𝑣₊^(𝑟) (4) is the distance between 𝑢 and 𝑣. It is proved that (ℝ_𝐹,𝐷) is a complete metric space with the properties [11, 12] (1)𝐷(𝑢⊕𝑤,𝑣⊕𝑤)=𝐷(𝑢,𝑣) for all 𝑢,𝑣,𝑤∈ℝ_𝐹,(2)𝐷(𝑘⊙𝑢,𝑘⊙𝑣)=|𝑘|𝐷(𝑢,𝑣) for all 𝑢,𝑣∈ℝ_𝐹 for all 𝑘∈ℝ,(3)𝐷(𝑢⊕𝑣,𝑤⊕𝑒)≤𝐷(𝑢,𝑤)+𝐷(𝑣,𝑒) for all 𝑢,𝑣,𝑤,𝑒∈ℝ_𝐹.

Definition 5 (see [10]). Let 𝑓,𝑔∶[𝑎,𝑏]→ℝ_𝐹 be fuzzy real number valued functions. The uniform distance between 𝑓, 𝑔 is defined by 𝐷^∗𝑓,𝑔=sup𝐷𝑓(𝑥),𝑔(𝑥)∣𝑥∈[𝑎,𝑏]. (5)

Definition 6 (see [10]). Let 𝑓∶[𝑎,𝑏]→ℝ_𝐹. 𝑓 is fuzzy-Riemann integrable to 𝐽∈ℝ_𝐹 if for any 𝜀>0, there exists 𝛿>0 such that for any division 𝑃={[𝑢,𝑣];𝜉} of [𝑎,𝑏] with the norms Δ(𝑝)<𝛿, one has 𝐷_𝑃^∗(𝑣−𝑢)⊙𝑓(𝜉),𝐼<𝜀, (6) where ∑^∗ denotes the fuzzy summation. In this case it is denoted by 𝐼=(𝐹𝑅)∫^𝑏_𝑎𝑓(𝑥)𝑑𝑥.

Theorem 7 (see [13]). If 𝑓,𝑔∶[𝑎,𝑏]⊆ℝ→ℝ_𝐹 are fuzzy continuous functions, then the function 𝐹∶[𝑎,𝑏]→ℝ₊ defined by 𝐹(𝑥)=𝐷(𝑓(𝑥),𝑔(𝑥)) is continuous on [𝑎,𝑏], and 𝐷(𝐹𝑅)^𝑏_𝑎𝑓(𝑥)𝑑𝑥,(𝐹𝑅)^𝑏_𝑎𝑔(𝑥)≤𝑑𝑥_𝑏𝑎𝐷𝑓(𝑥),𝑔(𝑥)𝑑𝑥. (7)

Definition 8 (see [14]). A fuzzy real number valued function 𝑓∶ℝ→ℝ_𝐹 is said to be continuous in 𝑥₀∈ℝ, if for each 𝜀>0 there is 𝛿>0 such that 𝐷(𝑓(𝑥),𝑓(𝑥₀))<𝜀, whenever 𝑥∈ℝ and |𝑥−𝑥₀|<𝛿. One says that 𝑓 is fuzzy continuous on ℝ if 𝑓 is continuous at each 𝑥₀∈ℝ and denotes the space of all such functions by 𝐶_𝐹(ℝ).

Definition 9 (see [14]). Let 𝑓∶ℝ→ℝ_𝐹. One calls 𝑓 a uniformly continuous fuzzy real number valued function, if and only if for any 𝜀>0 there exists 𝛿>0 whenever |𝑥−𝑦|≤𝛿; 𝑥,𝑦∈ℝ, implies that 𝐷(𝑓(𝑥),𝑓(𝑦))≤𝜀. One denotes it as 𝑓∈𝐶^𝑈_𝐹(ℝ). Also, one denotes the space of all fuzzy continuous functions 𝑓∶ℝ²→ℝ_𝐹 by 𝐶_𝐹(ℝ²).

Similarly we have the following.

Definition 10. Let 𝑓∶ℝ²→ℝ_𝐹. One calls 𝑓 a uniformly continuous fuzzy real number valued function, if and only if for any 𝜀>0 there exists 𝛿>0 whenever: (𝑥−𝑥₀)²+(𝑦−𝑦₀)²≤𝛿; 𝑥,𝑦,𝑥₀,𝑦₀∈ℝ, implies that 𝐷𝑓𝑥,𝑦,𝑓𝑥₀,𝑦₀≤𝜀. (8) One denotes it as 𝑓∈𝐶^𝑈_𝐹(ℝ²).

Definition 11 (see [14]). Let 𝑓∶ℝ→ℝ_𝐹, then 𝑓 is called a nondecreasing function if and only if whenever 𝑥₁≤𝑥₂, 𝑥₁,𝑥₂∈ℝ, one has that 𝑓(𝑥₁)≤𝑓(𝑥₂); that is, (𝑓(𝑥₁))₋^(𝑟)≤(𝑓(𝑥₂))₋^(𝑟) and (𝑓(𝑥₁))₊^(𝑟)≤(𝑓(𝑥₂))₊^(𝑟), for all 𝑟∈[0,1].

Definition 12 (see [10, 15]). Let 𝑓∶ℝ→ℝ_𝐹 be a bounded function, then function 𝜔_ℝ(𝑓,⋅)∶ℝ₊∪{0}→ℝ₊, 𝜔_ℝ𝑓,𝛿=sup𝐷𝑓(𝑥),𝑓𝑦∣𝑥,𝑦∈ℝ,𝑥−𝑦≤𝛿, (9) where ℝ₊ is the set of positive real numbers, is called the modulus of continuity of 𝑓 on ℝ.

For 𝑓∶ℝ²→ℝ_𝐹, the modulus of continuity 𝜔_ℝ²(𝑓,𝛿) is defined as follows: 𝜔_ℝ²𝑓,𝛿=sup𝐷𝑓(𝑢,𝑣),𝑓𝑥,𝑦∣(𝑢,𝑣),𝑥,𝑦∈ℝ²,(𝑢−𝑥)²+𝑣−𝑦²≤𝛿, (10) where 𝑓∈𝐶_𝐹(ℝ²) and 𝛿>0. Observe that, for all 𝑓∈𝐶_𝐹([𝑎,𝑏]²) and 𝜆,𝛿>0, one has 𝜔_[𝑎,𝑏]²𝑓,𝜆𝛿≤(1+[𝜆])𝜔𝑓,𝛿, (11) where [𝜆] is defined to be the greatest integer less than or equal to 𝜆.

Some properties of the modulus of continuity are presented below.

Theorem 13 (see [15]). The following properties hold: (1)𝐷(𝑓(𝑥),𝑓(𝑦))≤𝜔_[𝑎,𝑏](𝑓,|𝑥−𝑦|)  for any  𝑥,𝑦∈[𝑎,𝑏],(2)𝜔_[𝑎,𝑏](𝑓,𝛿)  is increasing  function of  𝛿,(3)𝜔_[𝑎,𝑏](𝑓,0)=0, (4)𝜔_[𝑎,𝑏](𝑓,𝛿₁+𝛿₂)≤𝜔_[𝑎,𝑏](𝑓,𝛿₁)+𝜔_[𝑎,𝑏](𝑓,𝛿₂)for any  𝛿₁,𝛿₂≥0,  and  𝑓∶ℝ→ℝ_𝐹,(5)𝜔_[𝑎,𝑏](𝑓,𝑛𝛿)≤𝑛𝜔_[𝑎,𝑏](𝑓,𝛿)  for any 𝛿≥0, 𝑛∈𝑁,  and 𝑓∶ℝ→ℝ_𝐹,(6)𝜔_[𝑎,𝑏](𝑓,𝜆𝛿)≤[𝜆]𝜔_[𝑎,𝑏](𝑓,𝛿)≤(𝜆+1)𝜔_[𝑎,𝑏](𝑓,𝛿)  for any 𝛿,𝜆≥0,  where  [⋅]  is the ceiling of the number, any 𝑓∶ℝ→ℝ_𝐹.(7)If  [𝑐,𝑑]⊆[𝑎,𝑏]  then 𝜔_[𝑐,𝑑](𝑓,𝛿)≤𝜔_[𝑎,𝑏](𝑓,𝛿).

In [1], the following theorem is proved.

Theorem 14. Let 𝑓∈𝐶_𝐹(ℝ) and the scaling function 𝜑(𝑥) a real-valued bounded function with supp𝜑(𝑥)⊆[−𝑎,𝑎], 0<𝑎<+∞, 𝜑(𝑥)≥0, such that ∑^∞_𝑗=−∞𝜑(𝑥−𝑗)=1 on ℝ. For 𝑘∈ℤ, 𝑥∈ℝ, put 𝐵_𝑘𝑓(𝑥)∶=^∞∗_𝑗=−∞𝑓𝑗2^𝑘⊙𝜑2^𝑘𝑥−𝑗, (12) which is a fuzzy-wavelet-like operator. Then 𝐷𝐵_𝑘𝑓(𝑥),𝑓(𝑥)≤𝜔_ℝ𝑎𝑓,2^𝑘,𝐷^∗𝐵_𝑘𝑓,𝑓≤𝜔_ℝ𝑎𝑓,2^𝑘, (13) for all 𝑥∈ℝ and 𝑘∈ℤ. If 𝑓∈𝐶^𝑈_𝐹(ℝ), then as 𝑘→+∞ one gets 𝜔_ℝ(𝑓,𝑎/2^𝑘)→0 and lim_𝑘→+∞𝐵_𝑘𝑓=𝑓, pointwise and uniformly with rates.

3. Main Results

Theorem 15. All assumption here are as in Theorem 14. Define for 𝑘∈ℤ, 𝑥∈ℝ, the fuzzy-wavelet-like operator 𝑅1_𝑘𝑓(𝑥)=^∞∗_𝑗=−∞2𝜑2^𝑘⊙()𝑥−𝑗𝐹𝑅₀^1/2𝑓𝑡2^𝑘+𝑗2^𝑘𝑑𝑡. (14) Then 𝐷𝑅1_𝑘𝑓(𝑥),𝑓(𝑥)≤2𝜔_ℝ𝑓,2𝑎+12^𝑘+1,𝐷^∗𝑅1_𝑘𝑓,𝑓≤2𝜔_ℝ𝑓,2𝑎+12^𝑘+1, (15) for all 𝑘∈ℤ, 𝑥∈ℝ. When 𝑓∈𝐶^𝑈_𝐹(ℝ) then as 𝑘→+∞ one gets 𝜔_ℝ(𝑓,(2𝑎+1)/2^𝑘+1)→0 and lim_𝑘→+∞𝑅1_𝑘𝑓=𝑓.

Proof. We want to estimate 𝐷𝑅1_𝑘𝑓(𝑥),𝑓(𝑥)=𝐷^∗_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}2𝜑2^𝑘𝑥−𝑗(𝐹𝑅)^{2𝜑(2𝑘𝑥−𝑗)(𝐹𝑅)}⊙₀^1/2𝑓𝑡2^𝑘+𝑗2^𝑘𝑑𝑡,𝑓(𝑥)⊙1=𝐷^∗_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}2𝜑2^𝑘⊙()𝑥−𝑗𝐹𝑅₀^1/2𝑓𝑡2^𝑘+𝑗2^𝑘𝑑𝑡,_{∗𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}2𝜑2^𝑘𝑥−𝑗⊙(𝐹𝑅)₀^1/2𝑓(𝑥)𝑑𝑡≤_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝐷2𝜑2^𝑘𝑥−𝑗⊙(𝐹𝑅)₀^1/2𝑓𝑡2^𝑘+𝑗2^𝑘2𝑑𝑡,2𝜑^𝑘𝑥−𝑗⊙(𝐹𝑅)₀^1/2𝑓(𝑥)𝑑𝑡=2_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝜑2^𝑘×𝑥−𝑗_01/2𝐷𝑓𝑡2^𝑘+𝑗2^𝑘,𝑓(𝑥)𝑑𝑡=∶(∗). (16) Clearly, we have −𝑎2^𝑘<𝑗2^𝑘𝑎−𝑥<2^𝑘⟶𝑡2^𝑘+𝑗2^𝑘−𝑥<2𝑎+12^𝑘+1,10<𝑡<2. (17) Therefore, we conclude that (∗)≤2_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝜑2^𝑘𝑥−𝑗𝜔_ℝ×𝑓,2𝑎+12^𝑘+1=2𝜔_ℝ𝑓,2𝑎+12^𝑘+1, (18) and hence 𝐷^∗𝑅1_𝐾𝑓,𝑓≤2𝜔_ℝ𝑓,2𝑎+12^𝑘+1. (19)

Theorem 16. All assumptions here are as in Theorem 14. Define for 𝑘∈ℤ, 𝑥∈ℝ, 𝑛≥2 the fuzzy-wavelet-like operator 𝑅2_𝑘𝑓(𝑥)=^∞∗_𝑗=−∞()×2⊙𝐹𝑅_01/2𝑓𝑡2^𝑘+𝑛+𝑗2^𝑘𝑑𝑡⊙𝜑2^𝑘𝑥−𝑗. (20) Then 𝐷𝑅2_𝑘𝑓(𝑥),𝑓(𝑥)≤2𝜔_ℝ2𝑓,^𝑛+1𝑎+12^𝑘+𝑛+1,𝐷^∗𝑅2_𝑘𝑓,𝑓≤2𝜔_ℝ2𝑓,^𝑛+1𝑎+12^𝑘+𝑛+1, (21) when 𝑓∈𝐶^𝑈_𝐹(ℝ) then as 𝑘→+∞ one gets 𝜔_ℝ(𝑓,(2^𝑛+1𝑎+1)/(2^𝑘+𝑛+1))→0 and lim_𝑘→+∞𝑅2_𝑘𝑓=𝑓.

Proof. We need to estimate 𝐷𝑅2_𝑘𝑓(𝑥),𝑓(𝑥)=𝐷^∗_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}2⊙(𝐹𝑅)₀^1/2𝑓𝑡2^𝑘+𝑛+𝑗2^𝑘2𝑑𝑡⊙𝜑^𝑘𝑥−𝑗,𝑓(𝑥)⊙1=𝐷^∗_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}()×2⊙𝐹𝑅_01/2𝑓𝑡2^𝑘+𝑛+𝑗2^𝑘𝑑𝑡⊙𝜑2^𝑘𝑥−𝑗_{∗𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]},2⊙(𝐹𝑅)₀^1/2𝑓(𝑥)𝑑𝑡⊙𝜑2^𝑘≤𝑥−𝑗_{∗𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}2𝐷(𝐹𝑅)₀^1/2𝑓𝑡2^𝑘+𝑛+𝑗2^𝑘𝑑𝑡⊙𝜑2^𝑘𝑥−𝑗_{∗𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]},(𝐹𝑅)₀^1/2𝑓(𝑥)𝑑𝑡⊙𝜑2^𝑘𝑥−𝑗≤2^∗_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝜑2^𝑘×𝑥−𝑗_01/2𝐷𝑓𝑡2^𝑘+𝑛+𝑗2^𝑘,𝑓(𝑥)𝑑𝑡=∶(∗∗). (22) Here, we have −𝑎2^𝑘<𝑗2^𝑘𝑎−𝑥<2^𝑘⟹𝑡2^𝑘+𝑗2^𝑘2−𝑥<^𝑛+1𝑎+12^𝑘+𝑛+1,10<𝑡<2. (23) Now, by using the above inequality in (∗∗), we can write (∗∗)≤2_{𝑗2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝜑2^𝑘𝑥−𝑗𝜔_ℝ×2𝑓,^𝑛+1𝑎+12^𝑘+𝑛+1=2𝜔_ℝ2𝑓,^𝑛+1𝑎+12^𝑘+𝑛+1, (24) and hence 𝐷^∗𝑅2_𝐾𝑓,𝑓≤2𝜔_ℝ2𝑓,^𝑛+1𝑎+12^𝑘+𝑛+1. (25)

Theorem 17. All assumptions here are as in Theorem 14. Define for 𝑘∈ℤ, 𝑥∈ℝ the fuzzy-wavelet-like operator 𝑅3_𝐾𝑓(𝑥)∶=^∞∗_𝑗=−∞𝛿_𝑘𝑗𝑓⊙𝜑2^𝑘𝑥−𝑗, (26) where 𝛿_𝑘𝑗𝑓∶=^𝑛𝑠̃^𝑛₌₀𝑟̃₌₀𝜔𝑟𝜔̃𝑠̃𝑗⊙𝑓2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘𝜔𝑟̃𝑠̃,𝑛∈ℕ,,𝜔≥0,^𝑛𝑟̃₌₀𝜔𝑟̃=1,^𝑛𝑠̃₌₀𝜔𝑠̃=1. (27) Then 𝐷𝑅3_𝑘𝑓(𝑥),𝑓(𝑥)≤𝜔_ℝ𝑓,𝑎+22^𝑘,𝐷^∗𝑅3_𝑘,𝑓≤𝜔_ℝ𝑓,𝑎+22^𝑘,𝑎𝑙𝑙𝑘∈ℤ,𝑥∈ℝ. (28) When 𝑓∈𝐶^𝑈_𝐹(ℝ) then as 𝑘→+∞ one gets 𝜔_ℝ(𝑓,(𝑎+2)/2^𝑘)→0 and lim_𝑘→+∞𝑅3_𝑘𝑓=𝑓.

Proof. We have the following: 𝐷^∞∗_𝑗=−∞𝛿_𝑘𝑗𝑓⊙𝜑2^𝑘𝑥−𝑗,𝑓(𝑥)=𝐷^∗_{2𝑘𝑥−𝑗∈[−𝑎,𝑎]}𝛿_𝑘𝑗𝑓⊙𝜑2^𝑘𝑥−𝑗_{∗2𝑘𝑥−𝑗∈[−𝑎,𝑎]},𝑓(𝑥)⊙𝜑2^𝑘𝑥−𝑗≤_𝑗𝜑2^𝑘𝑥−𝑗𝐷𝛿_𝑘𝑗𝑓,𝑓(𝑥)=_𝑗𝜑2^𝑘𝑥−𝑗×𝐷^𝑛𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠𝑓̃𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘_𝑛,𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃⊙𝑓(𝑥)≤_𝑗𝜑2^𝑘𝑥−𝑗^𝑛𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠𝐷̃×𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘,𝑓(𝑥)≤𝜔_ℝ𝑓,𝑎+22^𝑘. (29) Notice that −𝑎2^𝑘<𝑗2^𝑘𝑎−𝑥<2^𝑘𝑟̃,0<2^𝑘𝑛<𝑎2^𝑘,𝑠̃0<2^𝑘𝑛<12^𝑘. (30) Therefore, 𝑗2^𝑘+𝑟̃2^𝑘𝑛+𝑠̃2^𝑘𝑛−𝑥<𝑎+22^𝑘. (31) This completes the proof.

Theorem 18. Let 𝑓∈𝐶_𝐹(ℝ²) and the scaling function 𝜙𝑥,𝑦=𝜑(𝑥)𝜑𝑦 (32) a real-valued bounded function with supp𝜑(𝑥,𝑦)⊆[−𝑎,𝑎]×[−𝑏,𝑏],  0<𝑎, 𝑏<+∞,  𝜑(𝑥,𝑦)≥0, such that ∑^∞_𝑖=−∞∑^∞_𝑗=−∞𝜑(𝑥−𝑖)𝜑(𝑦−𝑗)=1 on ℝ². For 𝑘∈ℤ, 𝑥∈ℝ, put 2𝐷𝐺_𝐾𝑓𝑥,𝑦∶=^∞∗∞_𝑗=−∞^∗_𝑗=−∞𝑓𝑖2^𝑘,𝑗2^𝑘⊙𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗, (33) which is a fuzzy-wavelet-like operator. Then 𝐷2𝐷𝐺_𝑘𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,𝑎²+𝑏²2^𝑘,𝐷^∗2𝐷𝐺_𝑘𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,𝑎²+𝑏²2^𝑘, (34) for all 𝑥∈ℝ and 𝑘∈ℤ. If 𝑓∈𝐶^𝑈_𝐹(ℝ²), then as 𝑘→+∞𝑤 one gets 𝜔_ℝ²(𝑓,(𝑎²+𝑏²)/2^𝑘)→0 and lim_𝑘→+∞2𝐷𝐺_𝑘𝑓=𝑓, pointwise and uniformly with rates.

Proof. We have the following:𝐷2𝐷𝐺_𝐾𝑓𝑥,𝑦,𝑓𝑥,𝑦=𝐷^∞∗∞_𝑖=−∞^∗_𝑗=−∞𝑓𝑖2^𝑘,𝑗2^𝑘⊙𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗_{∞∗∞𝑖=−∞∗𝑗=−∞},𝜑2^𝑘𝑥−𝑖𝜑2^𝑘≤𝑦−𝑗⊙𝑓𝑥,𝑦_{∞∗∞𝑖=−∞∗𝑗=−∞}𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑓𝑖𝑦−𝑗×𝐷2^𝑘,𝑗2^𝑘,𝑓𝑥,𝑦=∶(∗∗∗). (35) According to this fact that 𝑖−𝑎2^𝑘<𝑥<𝑖+𝑎2^𝑘,𝑗−𝑏2^𝑘<𝑦<𝑗+𝑏2^𝑘, (36) we conclude that 𝑖2^𝑘≤𝑎−𝑥2^𝑘,𝑗2^𝑘≤𝑏−𝑦2^𝑘, (37) and hence (∗∗∗)≤_𝑖𝑗𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗𝜔_ℝ²×𝑓,𝑎²+𝑏²2^𝑘=𝜔_ℝ²𝑓,𝑎²+𝑏²2^𝑘,𝐷^∗2𝐷𝐺_𝐾𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,𝑎²+𝑏²2^𝑘. (38) This completes the proof.

Theorem 19. All assumptions here are as in Theorem 18. For 𝑘∈ℤ, 𝑥∈ℝ, put 2𝐷𝐺1_𝐾𝑓=𝑥,𝑦_{∞∗∞𝑖=−∞∗𝑗=−∞}2^2𝑘⊙()×𝐹𝑅_{2−𝑘02−𝑘0}𝑓𝑖𝑡+2^𝑘𝑗,𝑠+2^𝑘2𝑑𝑡𝑑𝑠⊙𝜑^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗, (39) which is a fuzzy-wavelet-like operator. Then 𝐷2𝐷𝐺1_𝑘𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘,𝐷^∗2𝐷𝐺1_𝑘𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘, (40) for all 𝑥∈ℝ and 𝑘∈ℤ. If 𝑓∈𝐶^𝑈_𝐹(ℝ²), then as 𝑘→+∞𝑤 one gets 𝜔_ℝ²𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘⟶0 (41) and lim_𝑘→+∞2𝐷𝐺1_𝑘𝑓=𝑓, pointwise and uniformly with rates.

Proof. We have the following:𝐷2𝐷𝐺1_𝐾𝑓𝑥,𝑦,𝑓𝑥,𝑦=𝐷^∞∗∞_𝑖=−∞^∗_𝑗=−∞2^2𝑘⊙(𝐹𝑅)^2−𝑘₀^2−𝑘₀𝑓𝑖𝑡+2^𝑘𝑗,𝑠+2^𝑘2𝑑𝑡𝑑𝑠⊙𝜑^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗_{∞∗∞𝑖=−∞∗𝑗=−∞},𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗2^2𝑘⊙(𝐹𝑅)^2−𝑘₀^2−𝑘₀𝑓𝑥,𝑦𝑑𝑡𝑑𝑠≤2^∞2𝑘∞_{𝑖=−∞𝑗=−∞}𝜑2^𝑘𝑥−𝑖𝜑2^𝑘)𝑦−𝑗×𝐷(𝐹𝑅^2−𝑘₀^2−𝑘₀𝑓𝑖𝑡+2^𝑘𝑗,𝑠+2^𝑘,(𝐹𝑅)^2−𝑘₀^2−𝑘₀𝑓𝑥,𝑦𝑑𝑡𝑑𝑠≤2^∞2𝑘∞_{𝑖=−∞𝑗=−∞}𝜑2^𝑘𝑥−𝑖𝜑2^𝑘×𝑦−𝑗_{2−𝑘02−𝑘0}𝐷𝑓𝑖𝑡+2^𝑘𝑗,𝑠+2^𝑘,𝑓𝑥,𝑦𝑑𝑡𝑑𝑠=∶(∗∗∗∗). (42) According to this fact that 10≤𝑡≤2^𝑘1,0≤𝑠≤2^𝑘,𝑖−𝑎2^𝑘<𝑥<𝑖+𝑎2^𝑘,𝑗−𝑏2^𝑘<𝑦<𝑗+𝑏2^𝑘, (43) we conclude that 𝑖𝑡+2^𝑘−𝑥≤𝑎+12^𝑘,𝑗𝑠+2^𝑘−𝑦≤𝑏+12^𝑘, (44) and hence (∗∗∗∗)≤2^2𝑘_𝑖𝑗𝜑2^𝑘𝑥−𝑖𝜑2^𝑘𝑦−𝑗𝜔_ℝ²×𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘2^−2𝑘=𝜔_ℝ²𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘,𝐷^∗2𝐷𝐺1_𝐾𝑓𝑥,𝑦,𝑓𝑥,𝑦≤𝜔_ℝ²𝑓,(𝑎+1)²+(𝑏+1)²2^𝑘. (45) This completes the proof.

Theorem 20. Let 𝑓∶ℝ→ℝ_𝐹 and the scaling function 𝜑(𝑥) a real-valued bounded function with supp𝜑⊆[−𝑎,𝑎], 0<𝑎<+∞, such that (1)∑^∞_𝑗=−∞𝜑(𝑥−𝑗)=1 on ℝ, (2)there exists 𝑎,𝑏∈ℝ such that 𝜑 is nondecreasing for 𝑥≤𝑎 and 𝜑 is nonincreasing for 𝑥≥𝑏 (the above imply 𝜑≥0). Let 𝑓(𝑥,𝑦) be nondecreasing fuzzy function, then (𝑅3_𝑘𝑓)(𝑥) is nondecreasing fuzzy valued functions for any 𝑘∈ℤ.

Proof. The proof is similar to the proof of [15, Theorem 12.14]. Let 𝑥_𝑛,𝑥∈ℝ such that 𝑥_𝑛→𝑥, as 𝑛→+∞, then 𝐷(𝑓(𝑥_𝑛),𝑓(𝑥))→0, by fuzzy continuity of 𝑓. But we have 𝐷𝑓𝑥_𝑛,𝑓(𝑥)=sup_[]𝑟∈0,1max𝑓𝑥_𝑛−^(𝑟)−𝑓(𝑥)₋^(𝑟),𝑓𝑥_𝑛+^(𝑟)−𝑓(𝑥)₊^(𝑟). (46) That is, |(𝑓(𝑥_𝑛))_±^(𝑟)−(𝑓(𝑥))_±^(𝑟)|→0, for all 0≤𝑟≤1 as 𝑛→+∞, respectively. Therefore (𝑓)_±^(𝑟)∈𝐶(ℝ), for all 0≤𝑟≤1, that is, real-valued continuous functions on ℝ. Since 𝑓 is fuzzy nondecreasing by Definition 11 we get that (𝑓)_±^(𝑟) are non-decreasing, for all 𝑟∈[0,1], respectively.
Next we observe that 𝑅3_𝑘𝑓(𝑥)^𝑟=^∞𝑛_𝑗=−∞𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃×𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘𝑟𝜑2^𝑘𝑥−𝑗. (47) That is, 𝑅3_𝑘𝑓(𝑥)₋^(𝑟),𝑅3_𝑘𝑓(𝑥)₊^(𝑟)=^∞_𝑗=−∞^𝑛𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘₋^(𝑟)_𝑛,𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘₊^(𝑟)×𝜑2^𝑘=𝑥−𝑗_{∞𝑛𝑗=−∞}𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘₋^(𝑟)×𝜑2^𝑘𝑥−𝑗_{∞𝑛𝑗=−∞},𝑠̃^𝑛₌₀𝑟̃₌₀𝑤𝑟𝑤̃𝑠̃𝑓𝑗2^𝑘+𝑟̃𝑛2^𝑘+𝑠̃𝑛2^𝑘₊^(𝑟)𝜑2^𝑘=𝐸𝑥−𝑗_𝑘𝑓₋^(𝑟)(𝑥),𝐸_𝑘𝑓₊^(𝑟)(𝑥). (48) So whenever 𝑥₁≤𝑥₂ we get 𝑅3_𝑘𝑓_±^(𝑟)𝑥₁≤𝑅3_𝑘𝑓_±^(𝑟)𝑥₂,∀𝑟∈[0,1]. (49) Therefore, 𝑅3_𝑘𝑓 is nondecreasing.

Theorem 21. Let 𝑓 and 𝜑 be as in Theorem 20. Then (𝑅1_𝑘𝑓)(𝑥) is a nondecreasing fuzzy valued function for any 𝑘∈ℤ.

Proof. The proof is similar to the proof of Theorem 20.

4. Numerical Examples

In this section, we present two examples. Also, we apply the following scaling function [1]: 𝜑(𝑥)=1−121≤𝑥≤2,0elsewhere. (50)

Example 1. In this example, we want to apply the fuzzy-wavelet-like operator 𝑅1_𝑘𝑓 defined in Theorem 15 to approximate a given function 𝑓(𝑥)=𝑒^𝑢𝑥, where [𝑢]^𝑟=[1+𝑟,3−𝑟], 𝑟∈[0,1]. To do this, we consider 𝑘=6. Comparing the numerical result with 𝑓(𝑥) is done in Figure 1.

Figure 1 

Comparing 𝑓(𝑥) and fuzzy-wavelet-like operator 𝑅1𝑘𝑓 with 𝑘=6  in 𝑥=0.5.

Example 2. Consider 𝑓(𝑥,𝑦)=𝑒^𝑢𝑥𝑦, where [𝑢]^𝑟=[𝑟,2−𝑟], 𝑟∈[0,1]. Now, we apply the fuzzy- wavelet-like operator 2𝐷𝐺_𝑘𝑓 defined in Theorem 18 to approximate 𝑓(𝑥,𝑦). We also compare the result of using 2𝐷𝐺_𝑘𝑓 with 𝑓(𝑥,𝑦) in 𝑥=0.9, 𝑦=0.9. For more details, see Figures 2 and 3.

Figure 2 

The fuzzy number 𝑓(𝑥,𝑦) in  𝑥=0.9, 𝑦=0.9.

Figure 3 

Approximation of 𝑓(𝑥,𝑦) by using fuzzy wavelet like operator 2𝐷𝐺𝑘𝑓 with 𝑘=10  in  𝑥=0.9, 𝑦=0.9.

5. 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 2𝐷𝐺_𝑘𝑓 defined in Theorem 18. Consider the following fuzzy integral: (𝐹𝑅)¹₋₁¹₋₁𝑓𝑥,𝑦𝑑𝑥𝑑𝑦. (51) By using the fuzzy-wavelet-like operator 2𝐷𝐺_𝑘𝑓, we get (𝐹𝑅)¹₋₁¹₋₁𝑓≅()𝑥,𝑦𝑑𝑥𝑑𝑦𝐹𝑅¹₋₁¹₋₁2𝐷𝐺_𝐾𝑓=()𝑥,𝑦𝑑𝑥𝑑𝑦𝐹𝑅¹₋₁^1∞₋₁^∗∞_𝑖=−∞^∗_𝑗=−∞𝑓𝑖2^𝑘,𝑗2^𝑘⊙𝜑2^𝑘𝑥−𝑖𝜑2^𝑘=𝑦−𝑗𝑑𝑥𝑑𝑦_{∞∗∞𝑖=−∞∗𝑗=−∞}𝑓𝑖2^𝑘,𝑗2^𝑘⊙¹₋₁𝜑2^𝑘𝑥−𝑖𝑑𝑥¹₋₁𝜑2^𝑘=𝑦−𝑗𝑑𝑦_{∞∗∞𝑖=−∞∗𝑗=−∞}𝑓𝑖2^𝑘,𝑗2^𝑘⊙12^{𝑘2𝑘−𝑖}₋₂^𝑘_−𝑖𝜑(𝑢)1𝑑𝑢2^{𝑘2𝑘−𝑗}₋₂^𝑘_−𝑗𝜑(𝑢)=𝑑𝑢_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗}𝑓𝑖2^𝑘,𝑗2^𝑘⊙12^𝑘1/2_−1/2𝜑(𝑢)𝑑𝑢²=12^2𝑘⊙^2𝑘−1_{𝑖=−2𝑘}^∗2𝑘−1_{𝑗=−2𝑘}^∗𝑓𝑖2^𝑘,𝑗2^𝑘. (52)

So, we have (𝐹𝑅)¹₋₁¹₋₁𝑓1𝑥,𝑦𝑑𝑥𝑑𝑦≅2^2𝑘⊙^2𝑘−1_{𝑖=−2𝑘}^∗2𝑘−1_{𝑗=−2𝑘}^∗𝑓𝑖2^𝑘,𝑗2^𝑘. (53)

Theorem 22. Let 𝑓∈𝐶_𝐹([𝑎,𝑏]²). Then 𝐷(𝐹𝑅)¹₋₁¹₋₁𝑓1𝑥,𝑦𝑑𝑥𝑑𝑦,2^2𝑘⊙^2𝑘−1_{𝑖=−2𝑘}^∗2𝑘−1_{𝑗=−2𝑘}^∗𝑖2^𝑘,𝑗2^𝑘≤4𝜔_[−1,1]²√𝑓,22^𝑘. (54)

Proof. We have the following:𝐷(𝐹𝑅)¹₋₁¹₋₁𝑓1𝑥,𝑦𝑑𝑥𝑑𝑦,2^2𝑘⊙^2𝑘−1_{𝑖=−2𝑘}^∗2𝑘−1_{𝑗=−2𝑘}^∗𝑓𝑖2^𝑘,𝑗2^𝑘=𝐷^2𝑘−1_{𝑖=−2𝑘}^∗2𝑘−1_{𝑗=−2𝑘}^∗(𝐹𝑅)^{(𝑗+1)/2𝑘}_𝑗/2𝑘^{(𝑖+1)/2𝑘}_𝑖/2𝑘𝑓𝑥,𝑦𝑑𝑥𝑑𝑦,_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗}(𝐹𝑅)^{(𝑗+1)/2𝑘}_𝑗/2𝑘^{(𝑖+1)/2𝑘}_𝑖/2𝑘𝑓𝑖2^𝑘,𝑗2^𝑘≤𝑑𝑥𝑑𝑦_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗(𝑗+1)/2𝑘𝑗/2𝑘(𝑖+1)/2𝑘𝑖/2𝑘}𝐷𝑓𝑖𝑥,𝑦,𝑓2^𝑘,𝑗2^𝑘≤𝑑𝑥𝑑𝑦_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗(𝑖+1)/2𝑘𝑖/2𝑘(𝑗+1)/2𝑘𝑗/2𝑘}𝜔_[−1,1]²𝑓,𝑖𝑥−2^𝑘2+𝑗𝑦−2^𝑘2≤𝑑𝑥𝑑𝑦_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗(𝑖+1)/2𝑘𝑖/2𝑘(𝑗+1)/2𝑘𝑗/2𝑘}𝜔_[−1,1]²√𝑓,22^𝑘=𝑑𝑥𝑑𝑦_{2𝑘−1𝑖=−2𝑘∗2𝑘−1𝑗=−2𝑘∗}12^2𝑘𝜔_[−1,1]²√𝑓,22^𝑘=4𝜔_[−1,1]²√𝑓,22^𝑘. (55)

Example 3. Consider the following fuzzy integral: (𝐹𝑅)¹₋₁¹₋₁𝑒^{𝑢𝑥−𝑦}𝑑𝑥𝑑𝑦, (56) where 𝑢=(1+𝑟,3−𝑟). The exact solution of this example is (𝐼₁(𝑟),𝐼₂(𝑟)), 0≤𝑟≤1, where 𝐼₁(𝑟)=1−𝑒^𝑟−3+13−𝑟𝑒𝑟+1^𝑟+1−1𝑒−𝑒⁻¹,𝐼₂(𝑟)=1−𝑒^−(𝑟+1)+1𝑟+1𝑒3−𝑟^3−𝑟−1𝑒−𝑒⁻¹. (57) Let 2𝐷𝐺_𝑘𝑓=(2𝐷𝐺_𝑘𝑓(𝑟),2𝐷𝐺_𝑘𝑓(𝑟)), 0≤𝑟≤1. By using proposed method in (53), we present approximate solution to this example for different values of 𝑘. To compare the exact and the approximate solutions, see Tables 1, 2, and 3.

Table 1 

Numerical results for Example 3 with 𝑘=6.

Table 2 

Numerical results for Example 3 with 𝑘=8.

Table 3 

Numerical results for Example 3 with 𝑘=10.

6. 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 2𝐷𝐺_𝑘𝑓 defined in Theorem 18, an application is presented.