# Some Fuzzy-Wavelet-Like Operators and Their Convergence

Authors: R. Ezzati; F. Mokhtarnejad; N. Hassasi
Journal: Mathematical Problems in Engineering
 (2013)

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## 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 operatorI. 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.


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## Body

## 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 operatorI 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 Section2, 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 functionu: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
(1)(i.e.,u(λx+(1-λ)y)≥min{u(x),u(y)}∀x,y∈ℝ,λ∈[0,1]),(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 byℝF.Definition 2 (see [9]).
Suppose thatu∈ℝF. 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 toℝF. In other words, for u,v∈ℝF, and λ∈ℝ, one defines uniquely the sum u⊕v and the product λ⊙u by
(2)[u⊕v]r=[u]r+[v]r,[λ⊙u]r=λ[u]r,∀r∈[0,1],
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 inℝF are defined as follows:  (1)
u⊕v=(u_(r)+v_(r),u¯(r)+v¯(r)),(2)
(3)(λ⊙u)={(λu_(r),λu¯(r))λ≥0,(λu¯(r),λu_(r))λ<0.Definition 4 (see [10]).
For arbitrary fuzzy numbersu, v, the quantity
(4)D(u,v)=supr∈[0,1]max{|u-(r)-v-(r)|,|u+(r)-v+(r)|}
is the distance between u and v. It is proved that (ℝF,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∈ℝF,(2)
D(k⊙u,k⊙v)=|k|D(u,v) for all u,v∈ℝF for all k∈ℝ,(3)
D(u⊕v,w⊕e)≤D(u,w)+D(v,e) for all u,v,w,e∈ℝF.Definition 5 (see [10]).
Letf,g:[a,b]→ℝF be fuzzy real number valued functions. The uniform distance between f, g is defined by
(5)D*(f,g)=sup{D(f(x),g(x))∣x∈[a,b]}.Definition 6 (see [10]).
Letf:[a,b]→ℝF. f is fuzzy-Riemann integrable to J∈ℝF 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
(6)D(∑P*(v-u)⊙f(ξ),I)<ε,
where ∑* denotes the fuzzy summation. In this case it is denoted by I=(FR)∫ab‍f(x)dx.Theorem 7 (see [13]).
Iff,g:[a,b]⊆ℝ→ℝF 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
(7)D((FR)∫ab‍f(x)dx,(FR)∫ab‍g(x)dx)≤∫ab‍D(f(x),g(x))dx.Definition 8 (see [14]).
A fuzzy real number valued functionf:ℝ→ℝF 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 CF(ℝ).Definition 9 (see [14]).
Letf:ℝ→ℝF. 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∈CFU(ℝ). Also, one denotes the space of all fuzzy continuous functions f:ℝ2→ℝF by CF(ℝ2).Similarly we have the following.Definition 10.
Letf:ℝ2→ℝF. 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
(8)D(f(x,y),f(x0,y0))≤ε.
One denotes it as f∈CFU(ℝ2).Definition 11 (see [14]).
Letf:ℝ→ℝF, 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]).
Letf:ℝ→ℝF be a bounded function, then function ωℝ(f,·):ℝ+∪{0}→ℝ+,
(9)ωℝ(f,δ)=sup{D(f(x),f(y))∣x,y∈ℝ,|x-y|≤δ},
where ℝ+ is the set of positive real numbers, is called the modulus of continuity of f on ℝ.Forf:ℝ2→ℝF, the modulus of continuity ωℝ2(f,δ) is defined as follows:
(10)ωℝ2(f,δ)=sup{(u-x)2+(v-y)2D(f(u,v),f(x,y))∣(u,v),(x,y)∈ℝ2,(u-x)2+(v-y)2≤δ},
where f∈CF(ℝ2) and δ>0. Observe that, for all f∈CF([a,b]2) and λ,δ>0, one has
(11)ω[a,b]2(f,λδ)≤(1+[λ])ω(f,δ),
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→ℝF,(5)
ω[a,b](f,nδ)≤nω[a,b](f,δ)  for anyδ≥0,  n∈N,  andf:ℝ→ℝF,(6)
ω[a,b](f,λδ)≤[λ]ω[a,b](f,δ)≤(λ+1)ω[a,b](f,δ)  for anyδ,λ≥0,  where  [·]  is the ceiling of the number, anyf:ℝ→ℝF.(7)
If  [c,d]⊆[a,b]  thenω[c,d](f,δ)≤ω[a,b](f,δ).In [1], the following theorem is proved.Theorem 14.
Letf∈CF(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
(12)(Bkf)(x):=∑j=-∞∞*f(j2k)⊙φ(2kx-j),
which is a fuzzy-wavelet-like operator. Then
(13)D((Bkf)(x),f(x))≤ωℝ(f,a2k),D*(Bkf,f)≤ωℝ(f,a2k),
for all x∈ℝ and k∈ℤ. If f∈CFU(ℝ), then as k→+∞ one gets ωℝ(f,a/2k)→0 and limk→+∞Bkf=f, pointwise and uniformly with rates.


## 3. Main Results

Theorem 15.
All assumption here are as in Theorem14. Define for k∈ℤ, x∈ℝ, the fuzzy-wavelet-like operator
(14)(R1kf)(x)=∑j=-∞∞*2φ(2kx-j)⊙(FR)∫01/2‍f(t2k+j2k)dt.
Then
(15)D((R1kf)(x),f(x))≤2ωℝ(f,2a+12k+1),D*((R1kf),f)≤2ωℝ(f,2a+12k+1),
for all k∈ℤ, x∈ℝ. When f∈CFU(ℝ) then as k→+∞ one gets ωℝ(f,(2a+1)/2k+1)→0 and limk→+∞R1kf=f.Proof.
We want to estimate(16)D((R1kf)(x),f(x))=D(∑j2kx-j∈[-a,a]2kx-j*2φ(2kx-j)(FR)2φ(2kx-j)(FR)⊙∫01/2‍f(t2k+j2k)dt,f(x)⊙1(∑j2kx-j∈[-a,a]‍*2φ(2kx-j)(FR))=D(∑j2kx-j∈[-a,a]2kx-j*2φ(2kx-j)⊙(FR)∫01/2‍f(t2k+j2k)dt,∑j2kx-j∈[-a,a]2kx-j*2φ(2kx-j))⊙(FR)∫01/2‍f(x)dt≤∑j2kx-j∈[-a,a]‍D(2φ(2kx-j)⊙(FR)∫01/2‍f(t2k+j2k)dt,2φ(2kx-j)⊙(FR)∫01/2‍f(x)dt)=2∑j2kx-j∈[-a,a]‍φ(2kx-j)×∫01/2‍D(f(t2k+j2k),f(x))dt=:(*).
Clearly, we have
(17)-a2k<j2k-x<a2k⟶t2k+j2k-x<2a+12k+1,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa0<t<12.
Therefore, we conclude that
(18)(*)≤2∑j2kx-j∈[-a,a]‍φ(2kx-j)ωℝ×(f,2a+12k+1)=2ωℝ(f,2a+12k+1),
and hence
(19)D*((R1Kf),f)≤2ωℝ(f,2a+12k+1).Theorem 16.
All assumptions here are as in Theorem14. Define for k∈ℤ, x∈ℝ, n≥2 the fuzzy-wavelet-like operator
(20)(R2kf)(x)=∑j=-∞∞*2⊙(FR)×∫01/2‍f(t2k+n+j2k)dt⊙φ(2kx-j).
Then
(21)D((R2kf)(x),f(x))≤2ωℝ(f,2n+1a+12k+n+1),D*((R2kf),f)≤2ωℝ(f,2n+1a+12k+n+1),
when f∈CFU(ℝ) then as k→+∞ one gets ωℝ(f,(2n+1a+1)/(2k+n+1))→0 and limk→+∞R2kf=f.Proof.
We need to estimate(22)D((R2kf)(x),f(x))=D(∑j2kx-j∈[-a,a]2kx-j*2⊙(FR)∫01/2‍f(t2k+n+j2k)dt⊙φ(2kx-j),f(x)⊙1∑j2kx-j∈[-a,a]*‍∑j2kx-j∈[-a,a]*‍)=D(∑j2kx-j∈[-a,a]2kx-j*2⊙(FR)×∫01/2‍f(t2k+n+j2k)dt⊙φ(2kx-j),∑j2kx-j∈[-a,a]2kx-j*2⊙(FR)∫01/2‍f(x)dt⊙φ(2kx-j)∑j2kx-j∈[-a,a]‍*2)≤∑j2kx-j∈[-a,a]2kx-j*2D(∑j2kx-j∈[-a,a]‍*2(FR)∫01/2‍f(t2k+n+j2k)dt⊙φ(2kx-j),∑j2kx-j∈[-a,a]2kx-j*(FR)∫01/2‍f(x)dt⊙φ(2kx-j)∑j2kx-j∈[-a,a]‍*2((FR)∫01/2‍f(t2k+n+j2k))≤2∑j2kx-j∈[-a,a]2kx-j*φ(2kx-j)×∫01/2‍D(f(t2k+n+j2k),f(x))dt=:(**).
Here, we have
(23)-a2k<j2k-x<a2k⟹t2k+j2k-x<2n+1a+12k+n+1,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa0<t<12.
Now, by using the above inequality in (**), we can write
(24)(**)≤2∑j2kx-j∈[-a,a]‍φ(2kx-j)ωℝ×(f,2n+1a+12k+n+1)=2ωℝ(f,2n+1a+12k+n+1),
and hence
(25)D*((R2Kf),f)≤2ωℝ(f,2n+1a+12k+n+1).Theorem 17.
All assumptions here are as in Theorem14. Define for k∈ℤ, x∈ℝ the fuzzy-wavelet-like operator
(26)(R3Kf)(x):=∑j=-∞∞*δkj(f)⊙φ(2kx-j),
where
(27)δkj(f)≔∑s~=0n‍∑r~=0n‍ωr~ωs~⊙f(j2k+r~n2k+s~n2k),n∈ℕ,ωr~,ωs~≥0,∑r~=0n‍ωr~=1,∑s~=0n‍ωs~=1.
Then
(28)D((R3kf)(x),f(x))≤ωℝ(f,a+22k),D*(R3k,f)≤ωℝ(f,a+22k),allk∈ℤ,x∈ℝ.
When f∈CFU(ℝ) then as k→+∞ one gets ωℝ(f,(a+2)/2k)→0 and limk→+∞R3kf=f.Proof.
We have the following:(29)D(∑j=-∞∞*δkj(f)⊙φ(2kx-j),f(x))=D(∑2kx-j∈[-a,a]2kx-j*δkj(f)⊙φ(2kx-j),∑2kx-j∈[-a,a]2kx-j*f(x)⊙φ(2kx-j))≤∑j‍φ(2kx-j)D(δkj(f),f(x))=∑j‍φ(2kx-j)×D(∑s~=0n‍∑r~=0n‍wr~ws~f(j2k+r~n2k+s~n2k),∑s~=0n‍∑r~=0n‍wr~ws~⊙f(x))≤∑j‍φ(2kx-j)∑s~=0n‍∑r~=0n‍wr~ws~D×(f(j2k+r~n2k+s~n2k),f(x))≤ωℝ(f,a+22k).
Notice that
(30)-a2k<j2k-x<a2k,0<r~2kn<a2k,0<s~2kn<12k.
Therefore,
(31)|j2k+r~2kn+s~2kn-x|<a+22k.
This completes the proof.Theorem 18.
Letf∈CF(ℝ2) and the scaling function
(32)ϕ(x,y)=φ(x)φ(y)
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
(33)(2DGKf)(x,y):=∑j=-∞∞*∑j=-∞∞*f(i2k,j2k)⊙φ(2kx-i)φ(2ky-j),
which is a fuzzy-wavelet-like operator. Then
(34)D((2DGkf)(x,y),f(x,y))≤ωℝ2(f,a2+b22k),D*(2DGkf(x,y),f(x,y))≤ωℝ2(f,a2+b22k),
for all x∈ℝ and k∈ℤ. If f∈CFU(ℝ2), then as k→+∞w one gets ωℝ2(f,(a2+b2)/2k)→0 and limk→+∞2DGkf=f, pointwise and uniformly with rates.Proof.
We have the following:(35)D((2DGKf)(x,y),f(x,y))=D(∑i=-∞∞*∑j=-∞∞*f(i2k,j2k)⊙φ(2kx-i)φ(2ky-j),∑i=-∞∞*∑j=-∞∞*φ(2kx-i)φ(2ky-j)⊙f(x,y))≤∑i=-∞∞*∑j=-∞∞*φ(2kx-i)φ(2ky-j)×D(f(i2k,j2k),f(x,y))=:(***).
According to this fact that
(36)i-a2k<x<i+a2k,j-b2k<y<j+b2k,
we conclude that
(37)|i2k-x|≤a2k,|j2k-y|≤b2k,
and hence
(38)(***)≤∑i‍∑j‍φ(2kx-i)φ(2ky-j)ωℝ2×(f,a2+b22k)=ωℝ2(f,a2+b22k),D*((2DGKf)(x,y),f(x,y))≤ωℝ2(f,a2+b22k).
This completes the proof.Theorem 19.
All assumptions here are as in Theorem18. For k∈ℤ, x∈ℝ, put
(39)(2DG1Kf)(x,y)=∑i=-∞∞*∑j=-∞∞*(⊙φ(2kx-i)φ(2ky-j)∑j=-∞∞)22k⊙(FR)×∫02-k‍∫02-k‍f(∑j=-∞∞t+i2k,s+j2k)dtds⊙φ(2kx-i)φ(2ky-j)∑j=-∞∞),
which is a fuzzy-wavelet-like operator. Then
(40)D((2DG1kf)(x,y),f(x,y))≤ωℝ2(f,(a+1)2+(b+1)22k),D*(2DG1kf(x,y),f(x,y))≤ωℝ2(f,(a+1)2+(b+1)22k),
for all x∈ℝ and k∈ℤ. If f∈CFU(ℝ2), then as k→+∞w one gets
(41)ωℝ2(f,(a+1)2+(b+1)22k)⟶0
and limk→+∞2DG1kf=f, pointwise and uniformly with rates.Proof.
We have the following:(42)D((2DG1Kf)(x,y),f(x,y))=D(∑i=-∞∞*∑j=-∞∞*22k⊙(FR)∫02-k‍∫02-k‍f(t+i2k,s+j2k)dtds⊙φ(2kx-i)φ(2ky-j),∑i=-∞∞*∑j=-∞∞*φ(2kx-i)φ(2ky-j)22k⊙(FR)∫02-k‍∫02-k‍f(x,y)dtds)≤22k∑i=-∞∞‍∑j=-∞∞‍φ(2kx-i)φ(2ky-j)×D((FR)∫02-k‍∫02-k‍f(t+i2k,s+j2k),(FR)∫02-k‍∫02-k‍f(x,y)dtds((FR)∫02-k‍∫02-k‍f(t+i2k,s+j2k),)≤22k∑i=-∞∞‍∑j=-∞∞‍φ(2kx-i)φ(2ky-j)×∫02-k‍∫02-k‍D(f(t+i2k,s+j2k),f(x,y)dtds(t+i2k,s+j2k))=:(****).
According to this fact that
(43)0≤t≤12k,0≤s≤12k,i-a2k<x<i+a2k,j-b2k<y<j+b2k,
we conclude that
(44)|t+i2k-x|≤a+12k,|s+j2k-y|≤b+12k,
and hence
(45)(****)≤22k∑i‍∑j‍φ(2kx-i)φ(2ky-j)ωℝ2×(f,(a+1)2+(b+1)22k)2-2k=ωℝ2(f,(a+1)2+(b+1)22k),D*((2DG1Kf)(x,y),f(x,y))≤ωℝ2(f,(a+1)2+(b+1)22k).
This completes the proof.Theorem 20.
Letf:ℝ→ℝF 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 existsa,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 (R3kf)(x) is nondecreasing fuzzy valued functions for any k∈ℤ.Proof.
The proof is similar to the proof of [15, Theorem 12.14]. Let xn,x∈ℝ such that xn→x, as n→+∞, then D(f(xn),f(x))→0, by fuzzy continuity of f. But we have
(46)D(f(xn),f(x))=supr∈[0,1]max{|(f(xn))-(r)-(f(x))-(r)|,|(f(xn))+(r)-(f(x))+(r)|}.
That is, |(f(xn))±(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(47)[(R3kf)(x)]r=∑j=-∞∞‍∑s~=0n‍∑r~=0n‍wr~ws~×[f(j2k+r~n2k+s~n2k)]rφ(2kx-j).
That is,
(48)[((R3kf)(x))-(r),((R3kf)(x))+(r)]=∑j=-∞∞‍(∑s~=0n‍∑r~=0n‍wr~ws~(f(j2k+r~n2k+s~n2k))-(r),∑s~=0n‍∑r~=0n‍wr~ws~(f(j2k+r~n2k+s~n2k))+(r))×φ(2kx-j)=∑j=-∞∞‍∑s~=0n‍∑r~=0n‍wr~ws~(f(j2k+r~n2k+s~n2k))-(r)×φ(2kx-j),∑j=-∞∞‍∑s~=0n‍∑r~=0n‍wr~ws~(f(j2k+r~n2k+s~n2k))+(r)φ(2kx-j)=[(Ek(f)-(r))(x),(Ek(f)+(r))(x)].
So whenever x1≤x2 we get
(49)(R3k(f)±(r))(x1)≤(R3k(f)±(r))(x2),∀r∈[0,1].
Therefore, R3kf is nondecreasing.Theorem 21.
Letf and φ be as in Theorem 20. Then (R1kf)(x) is a nondecreasing fuzzy valued function for any k∈ℤ.Proof.
The proof is similar to the proof of Theorem20.


## 4. Numerical Examples

In this section, we present two examples. Also, we apply the following scaling function [1]:
(50)φ(x)={1-12≤x≤12,0elsewhere.Example 1.
In this example, we want to apply the fuzzy-wavelet-like operatorR1kf defined in Theorem 15 to approximate a given function f(x)=eux, 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
Comparingf(x) and fuzzy-wavelet-like operator R1kf with k=6  in x=0.5.Example 2.
Considerf(x,y)=euxy, where [u]r=[r,2-r], r∈[0,1]. Now, we apply the fuzzy- wavelet-like operator 2DGkf defined in Theorem 18 to approximate f(x,y). We also compare the result of using 2DGkf with f(x,y) in x=0.9, y=0.9. For more details, see Figures 2 and 3.Figure 2
The fuzzy numberf(x,y) in  x=0.9, y=0.9.Figure 3
Approximation off(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 operator2DGkf defined in Theorem 18. Consider the following fuzzy integral:
(51)(FR)∫-11‍∫-11‍f(x,y)dxdy.
By using the fuzzy-wavelet-like operator 2DGkf, we get
(52)(FR)∫-11‍∫-11‍f(x,y)dxdy≅(FR)∫-11‍∫-11‍(2DGKf)(x,y)dxdy=(FR)∫-11‍∫-11‍∑i=-∞∞*∑j=-∞∞*f(i2k,j2k)⊙φ(2kx-i)φ(2ky-j)dxdy=∑i=-∞∞*∑j=-∞∞*f(i2k,j2k)⊙∫-11‍φ(2kx-i)dx∫-11‍φ(2ky-j)dy=∑i=-∞∞*∑j=-∞∞*f(i2k,j2k)⊙(12k∫-2k-i2k-i‍φ(u)du)(12k∫-2k-j2k-j‍φ(u)du)=∑i=-2k2k-1*∑j=-2k2k-1*f(i2k,j2k)⊙(12k∫-1/21/2‍φ(u)du)2=122k⊙∑i=-2k2k-1*∑j=-2k2k-1*f(i2k,j2k).So, we have(53)(FR)∫-11‍∫-11‍f(x,y)dxdy≅122k⊙∑i=-2k2k-1*∑j=-2k2k-1*f(i2k,j2k).Theorem 22.
Letf∈CF([a,b]2). Then
(54)D(∑i=-2k2k-1*‍(FR)∫-11‍∫-11‍f(x,y)dxdy,122k⊙∑i=-2k2k-1*∑j=-2k2k-1*(i2k,j2k))≤4ω[-1,1]2(f,22k).Proof.
We have the following:(55)D((FR)∫-11‍∫-11‍f(x,y)dxdy,122k⊙∑i=-2k2k-1*∑j=-2k2k-1*f(i2k,j2k))=D(∑i=-2k2k-1*∑j=-2k2k-1*(FR)∫j/2k(j+1)/2k‍∫i/2k(i+1)/2k‍f(x,y)dxdy,aaaaaaaaa∑i=-2k2k-1*∑j=-2k2k-1*(FR)∫j/2k(j+1)/2k‍∫i/2k(i+1)/2k‍f(i2k,j2k)dxdy)≤∑i=-2k2k-1*∑j=-2k2k-1*∫j/2k(j+1)/2k‍∫i/2k(i+1)/2k‍D(f(x,y),f(i2k,j2k))dxdy≤∑i=-2k2k-1*∑j=-2k2k-1*∫i/2k(i+1)/2k‍∫j/2k(j+1)/2k‍ω[-1,1]2(f,(x-i2k)2+(y-j2k)2)dxdy≤∑i=-2k2k-1*∑j=-2k2k-1*∫i/2k(i+1)/2k‍∫j/2k(j+1)/2k‍ω[-1,1]2(f,22k)dxdy=∑i=-2k2k-1*∑j=-2k2k-1*122kω[-1,1]2(f,22k)=4ω[-1,1]2(f,22k).Example 3.
Consider the following fuzzy integral:(56)(FR)∫-11‍∫-11‍eux-ydxdy,
where u=(1+r,3-r). The exact solution of this example is (I1(r),I2(r)), 0≤r≤1, where
(57)I1(r)=(1-er-33-r+1r+1(er+1-1))(e-e-1),I2(r)=(1-e-(r+1)r+1+13-r(e3-r-1))(e-e-1).
Let 2DGkf=(2DGkf_(r),2DGkf¯(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 Example3 with k=6.
r-level
|
2
D
G
k
f
_
-
I
1
|
|
2
D
G
k
f
¯
-
I
2
|
0.0
0.0570312
0.0892018
0.1
0.0556607
0.0704432
0.2
0.054025
0.0537745
0.3
0.0520895
0.0389791
0.4
0.0498158
0.0258617
0.5
0.0471608
0.0142472
0.6
0.0440764
0.0039777
0.7
0.0405092
0.0050884
0.8
0.0363992
0.0130785
0.9
0.0316798
0.020107
1.0
0.0262767
0.0262767Table 2
Numerical results for Example3 with k=8.
r-level
|
2
D
G
k
f
_
-
I
1
|
|
2
D
G
k
f
¯
-
I
2
|
0.0
0.0142312
0.0220079
0.1
0.0138981
0.0173406
0.2
0.0134993
0.0131953
0.3
0.0130261
0.00951781
0.4
0.012469
0.00625923
0.5
0.0118171
0.00337565
0.6
0.0110587
0.00082763
0.7
0.0101801
0.00142031
0.8
0.0916663
0.00340001
0.9
0.00800153
0.00514009
1.0
0.00666625
0.00666625Table 3
Numerical results for Example3 with k=10.
r-level
|
2
D
G
k
f
_
-
I
1
|
|
2
D
G
k
f
¯
-
I
2
|
0.0
0.00355617
0.00548333
0.1
0.0034735
0.00431794
0.2
0.00337443
0.00328303
0.3
0.0032568
0.00236503
0.4
0.00311822
0.000551771
0.5
0.00295602
0.0008321
0.6
0.00276719
0.000196327
0.7
0.00254839
0.000364475
0.8
0.0022959
0.000858269
0.9
0.00200556
0.00129221
1.0
0.00167272
0.00167272


## 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 operator2DGkf defined in Theorem 18, an application is presented.



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*Source: 832831-2013-07-11.xml*