research article on fourier series of fuzzy-valued functions · 2019. 7. 31. · on fourier series...

Research ArticleOn Fourier Series of Fuzzy-Valued Functions

ULur Kadak1,2 and Feyzi BaGar3

1 Department of Mathematics, Faculty of Science, Bozok University, Yozgat, Turkey2Department of Mathematics, Faculty of Science, Gazi University, Ankara, Turkey3 Department of Mathematics, Faculty of Arts and Sciences, Fatih University, 34500 İstanbul, Turkey

Correspondence should be addressed to Feyzi Başar; [email protected]

Received 13 November 2013; Accepted 30 December 2013; Published 10 April 2014

Academic Editors: A. Bellouquid, T. Calvo, and E. Momoniat

Copyright © 2014 U. Kadak and F. Başar. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in scienceand engineering. In the present paper since the utilization of Zadeh’s Extension principle is quite difficult in practice, we prefer theidea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We deriveuniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation andHenstock integration of a fuzzy-valued function with some necessary inclusions. Furthermore, Fourier series of periodic fuzzy-valued functions is defined and its complex form is given via sine and cosine fuzzy coefficients with an illustrative example. Finally,by using the Dirichlet kernel and its properties, we especially examine the convergence of Fourier series of fuzzy-valued functionsat each point of discontinuity, where one-sided limits exist.

1. Introduction

Fourier series were introduced by Joseph Fourier (1768–1830) for the purpose of solving the heat equation in ametal plate and it has long provided one of the principalmethods of analysis for mathematical physics, engineering,and signal processing. While the original theory of Fourierseries applies to the periodic functions occurring in wavemotion, such as with light and sound, its generalizations oftenrelate to wider settings, such as the time-frequency analysisunderlying the recent theories of wavelet analysis and localtrigonometric analysis. Additionally, the idea of Fourier wasto model a complicated heat source as a superposition (orlinear combination) of simple sine and cosine waves and towrite the solution as a superposition of the correspondingeigen solutions. This superposition or linear combination iscalled the Fourier series.

Due to the rapid development of the fuzzy theory,however, some of these basic concepts have been modifiedand improved. One of them set mapping operations to thecase of interval valued fuzzy sets. To accomplish this, weneed to introduce the idea of the level sets of interval fuzzy

sets and the related formulation of a representation of aninterval valued fuzzy set in terms of its level sets. Once havingthese structures, we can then provide the desired extension tointerval valued fuzzy sets.The effectiveness of level sets comesfrom not only their required memory capacity for fuzzy sets,but also from their two valued nature.This nature contributesto an effective derivation of the fuzzy-inference algorithmbased on the families of the level sets. Besides, the definitionof fuzzy sets by level sets offers advantages over membershipfunctions, especially when the fuzzy sets are in universes ofdiscourse with many elements.

Furthermore, we also study the Fourier series of periodicfuzzy-valued functions. Using a different approach, it canbe shown that the Fourier series with fuzzy coefficientsconverges. Applying this idea, we establish some connectionsbetween the Fourier series and Fourier series of fuzzy-valued functions with the level sets. Quite recently, by usingZadeh’s Extension Principle, M. Stojaković and Z. Stojakovićinvestigated the convergence of series of fuzzy numbers in[1] and they gave some results which complete their previousresults in [2]. Additionally, Talo and Başar [3] have extendedthe main results related to the sequence spaces and matrix

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 782652, 13 pageshttp://dx.doi.org/10.1155/2014/782652

2 The Scientific World Journal

transformations on the real or complex field to the fuzzynumbers with the level sets. Also, Kadak and Başar [4, 5]have recently studied the power series of fuzzy numbers andexamined on some sets of fuzzy-valued sequences with thelevel sets and gave some properties of the level sets togetherwith some inclusion relations in [6].

The rest of this paper is organized as follows. In Section 2,we give some required definitions and consequences relatedto the fuzzy numbers, sequences, and series of fuzzy numbers.We also report the most relevant and recent literature in thissection. In Section 3, first, the definition of periodic fuzzy-valued function is given which will be used in the proof ofour main results. In this section, Hukuhara differentiationand Henstock integration are presented according to fuzzy-valued functions which depend on 𝑥, 𝑡 ∈ [𝑎, 𝑏]. This sectionis terminated with the condensation of the results on uniformconvergence of fuzzy-valued sequences and series. In thefinal section of the paper, we assert that the Fourier seriesof a fuzzy-valued function with 2𝜋 period converges andespecially prove the convergence about a discontinuity pointby using Dirichlet kernel and one-sided limits.

2. Preliminaries, Background, and Notation

A fuzzy number is a fuzzy set on the real axis; that is, a map-ping 𝑢 : R → [0, 1] which satisfies the following four con-ditions.

(i) 𝑢 is normal; that is, there exists an 𝑥0∈ R such that

𝑢(𝑥0) = 1.

(ii) 𝑢 is fuzzy convex; that is, 𝑢[𝜆𝑥 + (1 − 𝜆)𝑦] ≥ min{𝑢(𝑥), 𝑢(𝑦)} for all 𝑥, 𝑦 ∈ R and for all 𝜆 ∈ [0, 1].

(iii) 𝑢 is upper semicontinuous.

(iv) The set [𝑢]0= {𝑥 ∈ R : 𝑢(𝑥) > 0} is compact (cf.

Zadeh [7]), where {𝑥 ∈ R : 𝑢(𝑥) > 0} denotes theclosure of the set {𝑥 ∈ R : 𝑢(𝑥) > 0} in the usualtopology of R.

Wedenote the set of all fuzzy numbers onR by𝐸1 and called itthe space of fuzzy numbers. 𝜆-level set [𝑢]

𝜆of 𝑢 ∈ 𝐸1 is defined

by

[𝑢]𝜆:=

{

{

{

{𝑡 ∈ R : 𝑢 (𝑡) ≥ 𝜆} , 0 < 𝜆 ≤ 1,

{𝑡 ∈ R : 𝑢 (𝑡) > 𝜆}, 𝜆 = 0.

(1)

The set [𝑢]𝜆is closed, bounded and, nonempty interval for

each 𝜆 ∈ [0, 1] which is defined by [𝑢]𝜆:= [𝑢−(𝜆), 𝑢+(𝜆)]. R

can be embedded in 𝐸1, since each 𝑟 ∈ R can be regarded asa fuzzy number 𝑟 defined by

𝑟 (𝑥) := {

1, 𝑥 = 𝑟,

0, 𝑥 ̸= 𝑟.

(2)

RepresentationTheorem (see [8]). Let [𝑢]𝜆= [𝑢−(𝜆), 𝑢+(𝜆)]

for 𝑢 ∈ 𝐸1 for each 𝜆 ∈ [0, 1]. Then the following statementshold.

(i) 𝑢− is a bounded and nondecreasing left continuousfunction on ]0, 1].

(ii) 𝑢+ is a bounded and nonincreasing left continuousfunction on ]0, 1].

(iii) The functions 𝑢− and 𝑢+ are right continuous at thepoint 𝜆 = 0.

(iv) 𝑢−(1) ≤ 𝑢+(1).

Conversely, if the pair of functions 𝑢− and 𝑢+ satisfies theconditions (i)–(iv), then there exists a unique 𝑢 ∈ 𝐸1 such that[𝑢]𝜆:= [𝑢−(𝜆), 𝑢+(𝜆)] for each 𝜆 ∈ [0, 1]. The fuzzy number 𝑢

corresponding to the pair of functions 𝑢− and 𝑢+ is defined by𝑢 : R → [0, 1], 𝑢(𝑥) := sup{𝜆 : 𝑢−(𝜆) ≤ 𝑥 ≤ 𝑢+(𝜆)}.

Definition 1 ((trapezoidal fuzzy number) [9, Definition, p.145]). We can define trapezoidal fuzzy number 𝑢 as 𝑢 = (𝑢

1,

𝑢2, 𝑢3, 𝑢4); the membership function 𝜇

(𝑢)of this fuzzy num-

ber will be interpreted as follows:

𝜇(𝑢) (

𝑥) :=

{{{{{{{

{{{{{{{

{

𝑥 − 𝑢1

𝑢2− 𝑢1

, 𝑢1≤ 𝑥 ≤ 𝑢

2,

1, 𝑢2≤ 𝑥 ≤ 𝑢

3,

𝑢4− 𝑥

𝑢4− 𝑢3

, 𝑢3≤ 𝑥 ≤ 𝑢

4,

0, 𝑢4< 𝑥 < 𝑢

1.

(3)

Then, the result [𝑢]𝜆:= [𝑢−(𝜆), 𝑢+(𝜆)] = [(𝑢

2− 𝑢1)𝜆 + 𝑢

1,

−(𝑢4− 𝑢3)𝜆 + 𝑢

4] holds for each 𝜆 ∈ [0, 1].

Let 𝑢, V, 𝑤 ∈ 𝐸1 and 𝛼 ∈ R. Then the operations addition,scalar multiplication and product defined on 𝐸1 by

𝑢 ⊕ V = 𝑤 ⇐⇒ [𝑤]𝜆 = [𝑢]𝜆 ⊕ [V]𝜆 ∀𝜆 ∈ [0, 1] ,

⇐⇒ 𝑤−(𝜆) = 𝑢

−(𝜆) + V− (𝜆) ,

𝑤+(𝜆) = 𝑢

+(𝜆) + V+ (𝜆) ∀𝜆 ∈ [0, 1] ,

[𝛼𝑢]𝜆= 𝛼[𝑢]𝜆

∀𝜆 ∈ [0, 1] ,

𝑢V = 𝑤 ⇐⇒ [𝑤]𝜆 = [𝑢]𝜆[V]𝜆 ∀𝜆 ∈ [0, 1] ,(4)

where it is immediate that

𝑤−(𝜆) = min {𝑢− (𝜆) V− (𝜆) , 𝑢− (𝜆) V+ (𝜆) ,

𝑢+(𝜆) V− (𝜆) , 𝑢+ (𝜆) V+ (𝜆)} ,

𝑤+(𝜆) = max {𝑢− (𝜆) V− (𝜆) , 𝑢− (𝜆) V+ (𝜆) ,

𝑢+(𝜆) V− (𝜆) , 𝑢+ (𝜆) V+ (𝜆)}

(5)

for all 𝜆 ∈ [0, 1]. Let 𝑊 be the set of all closed boundedintervals 𝐴 of real numbers with endpoints 𝐴 and 𝐴; that is,𝐴 := [𝐴, 𝐴]. Define the relation 𝑑 on𝑊 by

𝑑 (𝐴, 𝐵) := max {𝐴 − 𝐵

,

𝐴 − 𝐵

} . (6)

Then it can easily be observed that 𝑑 is a metric on 𝑊 (cf.Diamond and Kloeden [10]) and (𝑊, 𝑑) is a complete metric

The Scientific World Journal 3

space (cf. Nanda [11]). Now, we can define the metric𝐷 on 𝐸1by means of the Hausdorff metric 𝑑 as

𝐷 (𝑢, V) := sup𝜆∈[0,1]

𝑑 ([𝑢]𝜆, [V]𝜆)

:= sup𝜆∈[0,1]

max { 𝑢−(𝜆) − V− (𝜆)

,

𝑢+(𝜆) − V+ (𝜆)

} .

(7)

Definition 2 (see [12], Definition 2.1). 𝑢 ∈ 𝐸1 is said to bea nonnegative fuzzy number if and only if 𝑢(𝑥) = 0 for all𝑥 < 0. It is immediate that 𝑢 ⪰ 0 if 𝑢 is a nonnegative fuzzynumber.

One can see that

𝐷(𝑢, 0) = sup𝜆∈[0,1]

max {𝑢−(𝜆),𝑢+(𝜆)}

= max {𝑢−(0),𝑢+(0)} .

(8)

Proposition 3 (see [13]). Let 𝑢, V, 𝑤, 𝑧 ∈ 𝐸1 and 𝛼 ∈ R. Then,

(i) (𝐸1, 𝐷) is a completemetric space, (cf. Puri and Ralescu[14]).

(ii) 𝐷(𝛼𝑢, 𝛼V) = |𝛼|𝐷(𝑢, V).(iii) 𝐷(𝑢 ⊕ V, 𝑤 ⊕ V) = 𝐷(𝑢, 𝑤).(iv) 𝐷(𝑢 ⊕ V, 𝑤 ⊕ 𝑧) ≤ 𝐷(𝑢, 𝑤) + 𝐷(V, 𝑧).(v) |𝐷(𝑢, 0) − 𝐷(V, 0)| ≤ 𝐷(𝑢, V) ≤ 𝐷(𝑢, 0) + 𝐷(V, 0).

Definition 4. The following basic statements hold.

(i) [12, Definition 2.7] A sequence 𝑢 = (𝑢𝑘) of fuzzy

numbers is a function 𝑢 from the setN into the set 𝐸1.The fuzzy number𝑢

𝑘denotes the value of the function

at 𝑘 ∈ N and is called as the general term of thesequence. By 𝜔(𝐹), we denote the set of all sequencesof fuzzy numbers.

(ii) [12, Definition 2.9] A sequence (𝑢𝑛) ∈ 𝜔(𝐹) is called

convergent with limit 𝑢 ∈ 𝐸1 if and only if for every𝜀 > 0 there exists 𝑛

0= 𝑛0(𝜀) ∈ N such that𝐷(𝑢

𝑛, 𝑢) <

𝜀 for all 𝑛 ≥ 𝑛0.

(iii) [12, Definition 2.11] A sequence (𝑢𝑛) ∈ 𝜔(𝐹) is called

bounded if and only if the set of fuzzy numbersconsisting of the terms of the sequence (𝑢

𝑛) is a

bounded set; that is to say, a sequence (𝑢𝑛) ∈ 𝜔(𝐹) is

bounded if and only if there exist two fuzzy numbers𝑚 and𝑀 such that 𝑚 ⪯ 𝑢

𝑛⪯ 𝑀 for all 𝑛 ∈ N. This

means that 𝑚−(𝜆) ≤ 𝑢−𝑛(𝜆) ≤ 𝑀

−(𝜆) and 𝑚+(𝜆) ≤

𝑢+

𝑛(𝜆) ≤ 𝑀

+(𝜆) for all 𝜆 ∈ [0, 1].

Remark 5 (see [12]). According to Definition 4, the followingremarks can be given.

(a) Obviously the sequence (𝑢𝑛) ∈ 𝜔(𝐹) converges to a

fuzzy number 𝑢 if and only if {𝑢−𝑛(𝜆)} and {𝑢+

𝑛(𝜆)}

converge uniformly to 𝑢−(𝜆) and 𝑢+(𝜆) on [0, 1],respectively.

(b) The boundedness of the sequence (𝑢𝑛) ∈ 𝜔(𝐹) is

equivalent to the fact that

sup𝑛∈N

𝐷(𝑢𝑛, 0) = sup

𝑛∈N

sup𝜆∈[0,1]

max {𝑢−

𝑛(𝜆),𝑢+

𝑛(𝜆)} < ∞. (9)

If the sequence (𝑢𝑘) ∈ 𝜔(𝐹) is bounded then the

sequences of functions {𝑢−𝑘(𝜆)} and {𝑢+

𝑘(𝜆)} are uni-

formly bounded in [0, 1].

Definition 6 (see [12]). Let (𝑢𝑘) ∈ 𝜔(𝐹). Then the expression

⊕∑𝑘𝑢𝑘is called a series of fuzzy numbers with the level

summation⊕∑. Define the sequence (𝑠

𝑛) via 𝑛th partial level

sum of the series by

𝑠𝑛= 𝑢0⊕ 𝑢1⊕ 𝑢2⊕ ⋅ ⋅ ⋅ ⊕ 𝑢

𝑛 (10)

for all 𝑛 ∈ N. If the sequence (𝑠𝑛) converges to a fuzzy number

𝑢 then we say that the series⊕∑𝑘𝑢𝑘of fuzzy numbers

converges to 𝑢 and write⊕∑𝑘𝑢𝑘which implies that

lim𝑛→∞

𝑛

∑

𝑘=0

𝑢−

𝑘(𝜆) = 𝑢

−(𝜆) , lim

𝑛→∞

𝑛

∑

𝑘=0

𝑢+

𝑘(𝜆) = 𝑢

+(𝜆) ,

(11)

where the summation is in the sense of classical summationand converges uniformly in 𝜆 ∈ [0, 1]. Conversely, if

∑

𝑘

𝑢−

𝑘(𝜆) = 𝑢

−(𝜆) , ∑

𝑘

𝑢+

𝑘(𝜆) = 𝑢

+(𝜆) (12)

converge uniformly in 𝜆, then 𝑢 = {(𝑢−(𝜆), 𝑢+(𝜆)) : 𝜆 ∈ [0,1]} defines a fuzzy number such that 𝑢 =

⊕∑𝑘𝑢𝑘.

Definition 7 (see [12, Definition 2.14]). Let {𝑓𝑘(𝜆)} be a

sequence of functions defined on [𝑎, 𝑏] and 𝜆0∈ ]𝑎, 𝑏]. Then,

{𝑓𝑘(𝜆)} is said to be eventually equi-left-continuous at 𝜆

0if

for any 𝜀 > 0 there exist 𝑛0∈ N and 𝛿 > 0 such that

|𝑓𝑘(𝜆) − 𝑓

𝑘(𝜆0)| < 𝜀 whenever 𝜆 ∈ ]𝜆

0− 𝛿, 𝜆

0] and 𝑘 ≥ 𝑛

0.

Similarly, eventually equi-right-continuity at 𝜆0∈ [𝑎, 𝑏[ of

{𝑓𝑘(𝜆)} can be defined.

Theorem8 (see [12,Theorem2.15]). Let (𝑢𝑘) ∈ 𝜔(𝐹) such that

𝑢−

𝑘(𝜆) → 𝑢

−(𝜆) and 𝑢+

𝑘(𝜆) → 𝑢

+(𝜆), as 𝑘 → ∞ for each 𝜆 ∈

[0, 1]. Then, the pair of functions 𝑢− and 𝑢+ determines a fuzzynumber if and only if the sequences of functions {𝑢−

𝑘(𝜆)} and

{𝑢+

𝑘(𝜆)} are eventually equi-left-continuous at each 𝜆 ∈]0, 1]

and eventually equi-right-continuous at 𝜆 = 0.

Thus, it is deduced that the series∑∞𝑘=0𝑢−

𝑘(𝜆) = 𝑢

−(𝜆) and

∑∞

𝑘=0𝑢+

𝑘(𝜆) = 𝑢

+(𝜆) define a fuzzy number if the sequences

{𝑠−

𝑛(𝜆)} = {

𝑛

∑

𝑘=0

𝑢−

𝑘(𝜆)} , {𝑠

+

𝑛(𝜆)} = {

𝑛

∑

𝑘=0

𝑢+

𝑘(𝜆)} (13)

satisfy the conditions of Theorem 8.

Theorem 9 (cf. [13]). The following statements for level addi-tion ⊕ of fuzzy numbers and classical addition + of real scalarsare valid.


(i) 0 is neutral element with respect to ⊕; that is, 𝑢 ⊕ 0 =0 ⊕ 𝑢 = 𝑢 for all 𝑢 ∈ 𝐸1.

(ii) With respect to 0, none of 𝑢 ̸= 𝑟, 𝑟 ∈ R has opposite in𝐸1.

(iii) For any 𝛼, 𝛽 ∈ R with 𝛼, 𝛽 ≥ 0 or 𝛼, 𝛽 ≤ 0, and any𝑢 ∈ 𝐸1, we have (𝛼 + 𝛽)𝑢 = 𝛼𝑢 ⊕ 𝛽𝑢.

For general 𝛼, 𝛽 ∈ R, the above property does not hold.(iv) For any 𝛼 ∈ R and any 𝑢, V ∈ 𝐸1, we have 𝛼(𝑢 ⊕ V) =

𝛼𝑢 ⊕ 𝛼V.(v) For any 𝛼, 𝛽 ∈ R and any 𝑢 ∈ 𝐸1, we have 𝛼(𝛽𝑢) =

(𝛼𝛽)𝑢.

2.1. Generalized Hukuhara Difference. Let K be the spaceof nonempty compact and convex sets in the 𝑛-dimensionalEuclidean space R𝑛. If 𝑛 = 1, denote by 𝐼 the set of (closedbounded) intervals of the real line. Given two elements𝐴, 𝐵 ∈K and 𝛼 ∈ R, the usual interval arithmetic operations, thatis, addition and scalar multiplication, are defined by 𝐴 + 𝐵 ={𝑎+𝑏 : 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵} and𝛼𝐴 = {𝛼𝑎 : 𝑎 ∈ 𝐴}. It is well knownthat addition is associative and commutative andwith neutralelement {0}. If𝛼 = −1, scalarmultiplication gives the opposite−A = (−1)𝐴 = {−𝑎 : 𝑎 ∈ 𝐴} but, in general, 𝐴 + (−𝐴) ̸= 0;that is, the opposite of 𝐴 is not the inverse of 𝐴 in additionunless𝐴 is a singleton. A first consequence of this fact is that,in general, additive simplification is not valid.

To partially overcome this situation, theHukuhara differ-ence, H-difference for short, has been introduced as a set 𝐶for which 𝐴 ⊖ 𝐵 ⇔ 𝐴 = 𝐵 + 𝐶 and an important propertyof ⊖ is that 𝐴 ⊖ 𝐴 = {0} for all 𝐴 ∈ K and (𝐴 + 𝐵) ⊖ 𝐵 = 𝐴for all 𝐴, 𝐵 ∈ K. The H-difference is unique, but it does notalways exist. A necessary condition for 𝐴 ⊖ 𝐵 to exist is that𝐴 contains a translation {𝑐} + 𝐵 of 𝐵.

A generalization of the Hukuhara difference proposed in[15] aims to overcome this situation.

Definition 10 (see [15, Definition 1]). The generalized Huku-hara difference 𝐴 ⊖ 𝐵 of two sets 𝐴, 𝐵 ∈ K is defined asfollows:

𝐴 ⊖ 𝐵 = 𝐶 ⇐⇒ {

𝐴 = 𝐵 + 𝐶,

𝐵 = 𝐴 + (−1) 𝐶.

(14)

Proposition 11 (see [15]). The following statements hold.

(a) Let 𝐴, 𝐵 ∈ K be two compact convex sets. Then, wehave that

(i) if the H-difference exists, it is unique and is ageneralization of the usual Hukuhara differencesince 𝐴 ⊖ 𝐵 = 𝐴 − 𝐵, whenever 𝐴 ⊖ 𝐵 exists.

(ii) 𝐴 + (−𝐴) ̸= 0.(iii) (𝐴 + 𝐵) ⊖ 𝐵 = 𝐴.(iv) 𝐴 ⊖ 𝐵 = 𝐵 ⊖ 𝐴 = 𝐶 ⇔ 𝐶 = {0} and 𝐴 = 𝐵.

(b) TheH-difference of two intervals𝐴 = [𝑎−, 𝑎+] and 𝐵 =[𝑏−, 𝑏+] always exists and

[𝑎−, 𝑎+] ⊖ [𝑏

−, 𝑏+] = [𝑐

−, 𝑐+] , (15)

where 𝑐− = min{𝑎− − 𝑏−, 𝑎+ − 𝑏+}, 𝑐+ = max{𝑎− − 𝑏−, 𝑎+ − 𝑏+}which hold in Definition 10 are satisfied simultaneously if andonly if the two intervals have the same length and 𝑐− = 𝑐+.

Proposition 12 (see [16]). The following statements hold.

(a) If 𝐴 and 𝐵 are two closed intervals, then 𝐷(𝐴, 𝐵) = 𝐷(𝐴 ⊖ 𝐵, {0}).

(b) Let 𝑢 : [𝑎, 𝑏] → 𝐼 be such that 𝑢(𝑥) = [𝑢−(𝑥), 𝑢+(𝑥)].Then, we have

lim𝑥→𝑥0

𝑢 (𝑥) = ℓ ⇐⇒ lim𝑥→𝑥0

(𝑢 (𝑥) ⊖ ℓ) = {0} ,

lim𝑥→𝑥0

𝑢 (𝑥) = 𝑢 (𝑥0) ⇐⇒ lim

𝑥→𝑥0

(𝑢 (𝑥) ⊖ 𝑢 (𝑥0)) = {0} ,

(16)

where the limits are in the Hausdorff metric 𝑑 forintervals.

3. Fuzzy-Valued Functions with the Level Sets

In this chapter, we consider sequences and series offuzzy-valued function and develop uniform convergence,Hukuhara differentiation, and Henstock integration. In addi-tion, we present characterizations of uniform convergencesigns in sequences of fuzzy-valued functions.

Definition 13 (see [6]). Consider a function𝑓𝑡 from [𝑎, 𝑏] into𝐸1 with respect to a membership function 𝜇

𝑓𝑡 which is called

trapezoidal fuzzy number and is interpreted as follows:

𝜇𝑓𝑡 (𝑥) :=

{{{{{{{{{{{{{

{{{{{{{{{{{{{

{

𝑥 − 𝑓1(𝑡)

𝑓2 (𝑡) − 𝑓1 (

𝑡)

, 𝑓1(𝑡) ≤ 𝑥 ≤ 𝑓

2(𝑡) ,

1, 𝑓2 (𝑡) ≤ 𝑥 ≤ 𝑓3 (

𝑡) ,

𝑓4(𝑡) − 𝑥

𝑓4(𝑡) − 𝑓

3(𝑡)

, 𝑓3 (𝑡) ≤ 𝑥 ≤ 𝑓4 (

𝑡) ,

0, 𝑓4(𝑡) < 𝑥 < 𝑓

1(𝑡) .

(17)

Then, the membership function turns out to be 𝑓𝑡(𝑥) =[𝑓−

𝜆(𝑡), 𝑓+

𝜆(𝑡)] = [(𝑓

2(𝑡) − 𝑓

1(𝑡))𝜆 + 𝑓

1(𝑡), 𝑓4(𝑡) − (𝑓

4(𝑡) −

𝑓3(𝑡))𝜆] ∈ 𝐸

1 consisting of each of the functions 𝑓−𝜆, 𝑓+

𝜆

depending on 𝑡 ∈ [𝑎, 𝑏] for all 𝜆 ∈ [0, 1]. Then, the function𝑓𝑡 is said to be a fuzzy-valued function on [𝑎, 𝑏] for all 𝑥, 𝑡 ∈[𝑎, 𝑏].

Remark 14. The functions 𝑓𝑖with 𝑖 ∈ {1, 2, 3, 4} given in

Definition 13 are also defined for all 𝑡 ∈ [𝑎, 𝑏] as 𝑓𝑖(𝑡) = 𝑘,

where 𝑘 is any constant.


Now, following Kadak [17], we give the classical sets𝐶𝐹[𝑎, 𝑏] and 𝐵

𝐹[𝑎, 𝑏] consisting of the continuous and

bounded fuzzy-valued functions; that is,

𝐶𝐹 [𝑎, 𝑏] := {𝑓

𝑡| 𝑓𝑡: [𝑎, 𝑏] → 𝐸

1∋ 𝑓𝑡 continuous

fuzzy-valued function ∀𝑥, 𝑡 ∈ [𝑎, 𝑏]} ,

𝐵𝐹 [𝑎, 𝑏] := {𝑓

𝑡| 𝑓𝑡: [𝑎, 𝑏] → 𝐸

1∋ 𝑓𝑡 bounded

fuzzy-valued function ∀𝑥, 𝑡 ∈ [𝑎, 𝑏]} .

(18)

Obviously, from Representation Theorem, each of the func-tions 𝑓−

𝜆, 𝑓+

𝜆which depend on 𝑡 ∈ [𝑎, 𝑏] is left continuous on

𝜆 ∈ (0, 1] and right continuous at 𝜆 = 0. It was shown that𝐶𝐹[𝑎, 𝑏] and 𝐵

𝐹[𝑎, 𝑏] are complete with the metric𝐷

𝐹∞on 𝐸1

defined by means of the Hausdorff metric 𝑑 as

𝐷𝐹∞(𝑓𝑡, 𝑔𝑡) := sup𝑥∈[𝑎,𝑏]

{𝐷 (𝑓𝑡(𝑥) , 𝑔

𝑡(𝑥))}

= sup𝑥∈[𝑎,𝑏]

{ sup𝜆∈[0,1]

𝑑 ([𝑓𝑡(𝑥)]𝜆, [𝑔𝑡(𝑥)]𝜆)}

:= max{ sup𝜆∈[0,1]

sup𝑡∈[𝑎,𝑏]

𝑓−

𝜆(𝑡) − 𝑔

−

𝜆(𝑡),

sup𝜆∈[0,1]


𝑓+

𝜆(𝑡) − 𝑔

+

𝜆(𝑡)} ,

(19)

where 𝑓𝑡 = 𝑓𝑡(𝑥) and 𝑔𝑡 = 𝑔𝑡(𝑥) are the elements of the sets𝐶𝐹[𝑎, 𝑏] or 𝐵

𝐹[𝑎, 𝑏] with 𝑥, 𝑡 ∈ [𝑎, 𝑏].

3.1. Generalized Hukuhara Differentiation. The concept offuzzy differentiability comes from a generalization of theHukuhara difference for compact convex sets. We proveseveral properties of the derivative of fuzzy-valued functionsconsidered here. As a continuation of Hukuhara deriva-tives for real fuzzy-valued functions [18], we can define H-differentiation of a fuzzy-valued function 𝑓𝑡 with respect tolevel sets. For short, throughout the paper, we write𝐻 insteadof “Hukuhara sense.”

Definition 15. A fuzzy-valued function 𝑓𝑡 : [𝑎, 𝑏] → 𝐸1 issaid to be generalized H-differentiable with respect to thelevel sets at 𝑥, 𝑡 ∈ [𝑎, 𝑏] if

(1) (𝑓𝑡)(𝑥) ∈ 𝐸1 exists such that, for all ℎ > 0 sufficientlynear to 0, the H-difference 𝑓𝑡(𝑥 + ℎ) ⊖ 𝑓𝑡(𝑥) exists;then the H-derivative (𝑓𝑡)(𝑥) is given as follows:

(𝑓𝑡)

(𝑥) = limℎ→0

+

[

𝑓𝑡(𝑥 + ℎ) ⊖ 𝑓

𝑡(𝑥)

ℎ

]

𝜆

= [ limℎ→0

+

𝑓−

𝜆(𝑡 + ℎ) − 𝑓

−

𝜆(𝑡)

ℎ

,

limℎ→0

+

𝑓+

𝜆(𝑡 + ℎ) − 𝑓

+

𝜆(𝑡)

ℎ

]

= [(𝑓−

𝜆(𝑡))

, (𝑓+

𝜆(𝑡))

] ,

(20)

or (2) (𝑓𝑡)(𝑥) ∈ 𝐸1 exists such that, for all ℎ < 0 sufficientlynear to 0, the H-difference 𝑓𝑡(𝑥 + ℎ) ⊖ 𝑓𝑡(𝑥) exists;then the H-derivative (𝑓𝑡)(𝑥) is given as follows:

(𝑓𝑡)

(𝑥) = limℎ→0

−

[

𝑓𝑡(𝑥 + ℎ) ⊖ 𝑓

𝑡(𝑥)

ℎ

]

𝜆

= [ limℎ→0

−

𝑓−

𝜆(𝑡 + ℎ) − 𝑓

−

𝜆(𝑡)

ℎ

,

limℎ→0

−

𝑓+

𝜆(𝑡 + ℎ) − 𝑓

+

𝜆(𝑡)

ℎ

]

= [(𝑓−

𝜆)

(𝑡) , (𝑓+

𝜆)

(𝑡)]

(21)

for all 𝑥, 𝑡 ∈ [𝑎, 𝑏] and 𝜆 ∈ [0, 1].

From here, we remember that the H-derivative of 𝑓𝑡 at𝑥, 𝑡 ∈ [𝑎, 𝑏] depends on the value 𝑡 and the choice of aconstant 𝜆 ∈ [0, 1].

Corollary 16. A fuzzy-valued function𝑓𝑡 is H-differentiable ifand only if 𝑓−

𝜆and 𝑓+

𝜆are differentiable functions in the usual

sense.

Definition 17 (periodicity). A fuzzy-valued function 𝑓𝑡 iscalled periodic if there exists a constant 𝑃 > 0 for which𝑓𝑡(𝑥 + 𝑃) = 𝑓

𝑡(𝑥) for any 𝑥, 𝑡 ∈ [𝑎, 𝑏]. Thus, it can easily

be seen that the conditions 𝑓−𝜆(𝑡+𝑃) = 𝑓

−

𝜆(𝑡) and 𝑓+

𝜆(𝑡+𝑃) =

𝑓+

𝜆(𝑡) hold for all 𝑡 ∈ [𝑎, 𝑏] and 𝜆 ∈ [0, 1]. Such a constant

𝑃 > 0 is called a period of the function 𝑓𝑡.

3.2. Generalized Fuzzy-Henstock Integration

Definition 18 (see [19, Definition 8.7]). A fuzzy valued func-tion 𝑓𝑡 is said to be fuzzy-Henstock, in short FH-integrable,if for any 𝜖 > 0, there exists 𝛿 > 0 such that

𝐷(∑

𝑃

(V − 𝑢) 𝑓𝑡 (𝜉) , 𝐼)

= sup𝜆∈[0,1]

max{

∑

𝑃

(V − 𝑢) 𝑓−𝜆(𝑡) − 𝐼

−

𝜆

,

∑

𝑃

(V − 𝑢) 𝑓+𝜆(𝑡) − 𝐼

+

𝜆

} < 𝜖

(22)

for any division𝑃 = {[𝑢, V]; 𝜉} of [𝑎, 𝑏]with the normsΔ(𝑃) <𝛿, where 𝐼 := (FH) ∫𝑏

𝑎𝑓𝑡(𝑥)𝑑𝑥 and 𝑡 ∈ [𝑎, 𝑏], and 𝑓𝑡 is FH-

integrable. One can conclude that ∑𝑃in (22) denotes the

usual Riemann sum for any division 𝑃 of [𝑎, 𝑏].

Theorem 19 (see [19, Theorem 8.8]). Let 𝑓𝑡 ∈ 𝐶𝐹[𝑎, 𝑏] and

FH-integrable on [𝑎, 𝑏]. If there exists 𝑥0∈ [𝑎, 𝑏] such that

𝑓−

𝜆(𝑥0) = 𝑓+

𝜆(𝑥0) = 1, then

[(FH) ∫𝑏

𝑎

𝑓𝑡(𝑥)𝑑𝑥]

𝜆

= [∫

𝑥0

𝑎

𝑓−

𝜆(𝑡) 𝑑𝑡, ∫

𝑏

𝑥0

𝑓+

𝜆(𝑡) 𝑑𝑡] .

(23)


Remark 20. We remark that the integrals ∫𝑏𝑎𝑓±

𝜆(𝑡)𝑑𝑡 in (23)

exist in the usual sense for all 𝜆 ∈ [0, 1] and 𝑡 ∈ [𝑎, 𝑏]. It iseasy to see that the pair of functions 𝑓±

𝜆: [𝑎, 𝑏] → R are

continuous.

Remark 21. Note that if 𝑓𝑡 is periodic fuzzy-valued functionand FH-integrable on any interval of length 𝑃, then it is FH-integrable on any other of the same length, and the value ofthe integral is the same; that is,

[(FH) ∫𝑎+𝑃

𝑎


𝜆

= [(FH) ∫𝑏+𝑃

𝑏


𝜆

(24)

for all 𝑥, 𝑡 ∈ [𝑎, 𝑏] and 𝜆 ∈ [0, 1].

This property is an immediate consequence of the inter-pretation of an integral as an area. In fact, each integral (24)equals the area bounded by the curves𝑓±(𝑡), the straight lines𝑥 = 𝑎 and 𝑥 = 𝑏, and the closed interval [𝑎, 𝑏] of 𝑥-axis. Inthe present case, the areas represented by two integrals arethe same because of the periodicity of 𝑓𝑡. Hereafter, whenwe say that a fuzzy-valued function 𝑓𝑡 with period 𝑃 is FH-integrable, we mean that it is FH-integrable on an interval oflength 𝑃. It follows from the property just proved that 𝑓𝑡 isalso FH-integrable on any interval of finite length.

Definition 22 (see [6] (uniform convergence)). Let {𝑓𝑡𝑛(𝑥)} be

a sequence of fuzzy-valued functions defined on a set 𝐴 ⊆R. We say that {𝑓𝑡

𝑛(𝑥)} converges pointwise on 𝐴 if for each

𝑥 ∈ 𝐴 the sequence {𝑓𝑡𝑛(𝑥)} converges for all 𝑥, 𝑡 ∈ 𝐴 and

𝜆 ∈ [0, 1]. If a sequence {𝑓𝑡𝑛(𝑥)} converges pointwise on a set

𝐴, then we can define 𝑓𝑡 : 𝐴 → 𝐸1 by

lim𝑛→∞

𝑓𝑡

𝑛(𝑥) = 𝑓

𝑡(𝑥) for each 𝑥, 𝑡 ∈ 𝐴. (25)

In other words, {𝑓𝑡𝑛(𝑥)} converges to 𝑓𝑡 on 𝐴 if and only

if for each 𝑥 ∈ 𝐴 and for an arbitrary 𝜖 > 0, there existsan integer 𝑁 = 𝑁(𝜖, 𝑥) such that 𝐷(𝑓𝑡

𝑛(𝑥), 𝑓

𝑡(𝑥)) < 𝜖

whenever 𝑛 > 𝑁.The integer𝑁 in the definition of pointwiseconvergence may, in general, depend on both 𝜖 > 0 and𝑥 ∈ 𝐴. If, however, one integer can be found that works for allpoints in 𝐴, then the convergence is said to be uniform. Thatis, a sequence of fuzzy-valued functions {𝑓𝑡

𝑛(𝑥)} converges

uniformly to 𝑓𝑡 on a set 𝐴 if, for each 𝜖 > 0, there exists aninteger𝑁(𝜖) such that

𝐷(𝑓𝑡

𝑛(𝑥) , 𝑓

𝑡(𝑥)) < 𝜖 whenever 𝑛 > 𝑁 (𝜖) , ∀𝑥, 𝑡 ∈ 𝐴.

(26)

Obviously, the sequence (𝑓𝑡𝑛) of fuzzy-valued functions con-

verges to a fuzzy valued-function 𝑓𝑡 if and only if {(𝑓−𝜆)𝑛(𝑡)}

and {(𝑓+𝜆)𝑛(𝑡)} converge uniformly to 𝑓−


𝜆(𝑡) in 𝜆 ∈

[0, 1], respectively. Often, we say that 𝑓𝑡 is the uniform limitof the sequence {𝑓𝑡

𝑛(𝑥)} on 𝐴 and write 𝑓𝑡

𝑛→ 𝑓𝑡, 𝑛 → ∞,

uniformly on 𝐴.Now, as a consequence of Definition 22, the following

theorem determines the characterization of uniform conver-gence of fuzzy-valued sequences.

Theorem 23 (see [6]). Let 𝑥, 𝑡 ∈ 𝐴 and 𝜆 ∈ [0, 1]. Then, thefollowing statements are valid.

(i) A sequence of fuzzy-valued functions {𝑓𝑡𝑛(𝑥)} defined

on a set 𝐴 ⊆ R converges uniformly to a fuzzy-valuedfunction 𝑓𝑡 on 𝐴 if and only if

𝛿𝑛= sup𝑥∈[𝑎,𝑏]

𝐷(𝑓𝑡

𝑛(𝑥) , 𝑓

𝑡(𝑥))

= sup𝑥∈[𝑎,𝑏]

{ sup𝜆∈[0,1]

𝑑 ([𝑓𝑡

𝑛(𝑥)]𝜆, [𝑓𝑡(𝑥)]𝜆)}

with lim𝑛→∞

𝛿𝑛= 0.

(27)

(ii) The limit of a uniformly convergent sequence of con-tinuous fuzzy-valued functions {𝑓𝑡

𝑛} on a set 𝐴 is

continuous. That is, for each 𝑎 ∈ 𝐴,

lim𝑥→𝑎

[ lim𝑛→∞

𝑓𝑡

𝑛(𝑥)] = lim

𝑛→∞[ lim𝑥→𝑎

𝑓𝑡

𝑛(𝑥)] . (28)

Theorem 24 (interchange of limit and integration). Supposethat 𝑓𝑡

𝑛(𝑥) ∈ 𝐶

𝐹[𝑎, 𝑏] for all 𝑛 ∈ N such that {𝑓𝑡

𝑛(𝑥)}

converges uniformly to 𝑓𝑡(𝑥) on [𝑎, 𝑏]. By combining this andthe inclusion (28), the equalities

lim𝑛→∞

[(FH) ∫𝑏

𝑎

𝑓𝑡

𝑛(𝑥)𝑑𝑥]

𝜆

= [(FH) ∫𝑏

𝑎

lim𝑛→∞

𝑓𝑡

𝑛(𝑥)𝑑𝑥]

𝜆

= [(FH) ∫𝑏

𝑎


𝜆

(29)

hold, where the integral (FH) ∫𝑏𝑎𝑓𝑡(𝑥)𝑑𝑥 exists for all 𝑥, 𝑡 ∈

[𝑎, 𝑏] and 𝜆 ∈ [0, 1]. Also, for each 𝑝 ∈ [𝑎, 𝑏], it is trivial that

lim𝑛→∞

[(FH) ∫𝑝

𝑎

𝑓𝑡

𝑛(𝑥)𝑑𝑥]

𝜆

= [(FH) ∫𝑝

𝑎


𝜆

= [∫

𝑝

𝑎

𝑓−


𝑝

𝑎

𝑓+

𝜆(𝑡) 𝑑𝑡]

(30)

and the convergence is uniform on [𝑎, 𝑏].

Proof. Note that by Part (ii) of Theorem 23, 𝑓𝑡 is continuouson [𝑎, 𝑏], so that (FH) ∫𝑏

𝑎𝑓𝑡(𝑥)𝑑𝑥 exists. Let 𝜀 > 0 be given.

Then, since 𝑓𝑡𝑛→ 𝑓𝑡 uniformly on [𝑎, 𝑏], there is an integer

𝑁 = 𝑁(𝜀) such that

𝐷[𝑓𝑡

𝑛(𝑥) , 𝑓

𝑡(𝑥)]

= max{ sup𝜆∈[0,1]


𝑓𝑛(𝑡)−

𝜆− 𝑓−

𝜆(𝑡),

sup𝜆∈[0,1]


𝑓𝑛(𝑡)+

𝜆− 𝑓+

𝜆(𝑡)} <

𝜀

(𝑏 − 𝑎)

(31)


for 𝑛 > 𝑁(𝜀). Again, since the distance function 𝐷(𝑓𝑡𝑛, 𝑓𝑡) is

continuous on [𝑎, 𝑏], it follows

𝐷[(FH) ∫𝑏

𝑎

𝑓𝑡

𝑛(𝑥) 𝑑𝑥, (FH) ∫

𝑏

𝑎

𝑓𝑡(𝑥) 𝑑𝑥]

= sup𝜆∈[0,1]

𝑑([(FH) ∫𝑏

𝑎

𝑓𝑡

𝑛(𝑥)𝑑𝑥]

𝜆

,

[(FH) ∫𝑏

𝑎


𝜆

)

(32)

and the equality on rigt-hand side in (32) is evaluated as

sup𝜆∈[0,1]

max{

∫

𝑏

𝑎

[𝑓𝑛(𝑡)−

𝜆− 𝑓(𝑡)

−

𝜆] 𝑑𝑡

,

∫

𝑏

𝑎

[𝑓𝑛(𝑡)+

𝜆− 𝑓(𝑡)

+

𝜆] 𝑑𝑡

}

≤ ∫

𝑏

𝑎

max{ sup𝜆∈[0,1]


𝑓𝑛(𝑡)−

𝜆− 𝑓(𝑡)

−

𝜆

,

sup𝜆∈[0,1]


𝑓𝑛(𝑡)+

𝜆− 𝑓(𝑡)

+

𝜆

} 𝑑𝑡

<

𝜀

𝑏 − 𝑎

(𝑏 − 𝑎) = 𝜀

(33)

for 𝑛 > 𝑁(𝜀). Since 𝜀 is arbitrary, this step completes the proof.

The hypothesis of Theorem 24 is sufficient for our pur-poses and may be used to show the nonuniform convergenceof the sequence {𝑓𝑡

𝑛(𝑥)} on [𝑎, 𝑏]. Also, it is important to point

out that a direct analogue of Theorem 24 for H-derivatives isnot true.

Remark 25. The uniform convergence of {𝑓𝑡𝑛(𝑥)} is sufficient

but is not necessary. In other words the conclusion of Theo-rem 24 holds without {𝑓𝑡

𝑛(𝑥)} being convergent uniformly on

[𝑎, 𝑏].

Definition 26. The series⊕∑∞

𝑘=1𝑓𝑡

𝑘(𝑥) is said to be uniformly

convergent to a fuzzy-valued function 𝑓𝑡(𝑥) on 𝐴 if thepartial level sum {𝑆𝑡

𝑛(𝑥)} converges uniformly to 𝑓𝑡(𝑥) on 𝐴.

That is, the series converges uniformly to 𝑓𝑡(𝑥) if, given any𝜀 > 0, there exists an integer 𝑛

0(𝜀) such that

𝐷[

⊕

∞

∑

𝑘=1

𝑓𝑡

𝑘(𝑥) , 𝑓

𝑡(𝑥)]

= max{ sup𝜆∈[0,1]


∞

∑

𝑘=1

𝑓𝑘(𝑡)−

𝜆− 𝑓(𝑡)

−

𝜆

,

sup𝜆∈[0,1]


∞

∑

𝑘=1

𝑓𝑘(𝑡)+

𝜆− 𝑓(𝑡)

+

𝜆

} < 𝜀

(34)

for all 𝑥, 𝑡 ∈ 𝐴 and 𝜆 ∈ [0, 1] whenever 𝑛 ≥ 𝑛0(𝜀).

Corollary 27. If {𝑓𝑡𝑘(𝑥)} is a continuous fuzzy-valued function

on 𝐴 ⊆ R for each 𝑘 ≥ 1 and⊕∑𝑘≥1𝑓t𝑘(𝑥) is uniformly conver-

gent to 𝑓𝑡(𝑥) on 𝐴, then 𝑓𝑡 is continuous on 𝐴 for all 𝑥, 𝑡 ∈ 𝐴.

Corollary 28 (interchange of summation and integra-tion). Suppose that {𝑓𝑡

𝑘(𝑥)} is a sequence in 𝐶

𝐹[𝑎, 𝑏] and

⊕∑∞

𝑘=0𝑓𝑡

𝑘(𝑥) converges uniformly to 𝑓𝑡(𝑥) on [𝑎, 𝑏]. Then,

[(FH)⊕

∞

∑

𝑘=0

∫

𝑏

𝑎

𝑓𝑡

𝑘(𝑥)𝑑𝑥]

𝜆

= [(FH) ∫𝑏

𝑎⊕

∞

∑

𝑘=0


𝜆

= [(FH) ∫𝑏

𝑎

𝑓𝑡

𝑘(𝑥)𝑑𝑥]

𝜆

,

(35)

where (FH) ∫𝑏𝑎𝑓𝑡(𝑥)𝑑𝑥 exists for all 𝑥, 𝑡 ∈ [𝑎, 𝑏] and 𝜆 ∈ [0, 1].

Now, we give an important trigonometric system whosespecial case of one of the systems of functions is applying tothe well-known inequalities.

By a trigonometric system we mean the system of 2𝜋periodic 𝑐𝑜𝑠𝑖𝑛𝑒 and 𝑠𝑖𝑛𝑒 functions which is given by

1, cos (𝑥) , sin (𝑥) , cos (2𝑥) , sin (2𝑥) , . . . ,

cos (𝑛𝑥) , sin (𝑛𝑥) , . . . ,(36)

for all 𝑛 ∈ N. We now prove some auxiliary formulas for anypositive integer 𝑛 such that ∫𝜋

−𝜋cos(𝑛𝑥)𝑑𝑥 = ∫𝜋

−𝜋sin(𝑛𝑥)𝑑𝑥 =

0. Therefore, one can see by using trigonometric identitiesthat

∫

𝜋

−𝜋

cos𝑚𝑥 cos 𝑛𝑥 𝑑𝑥 ={{

{{

{

0, 𝑚 ̸= 𝑛,

2𝜋, 𝑚 = 𝑛 = 0,

𝜋, 𝑚 = 𝑛 ̸= 0,

∫

𝜋

−𝜋

sin𝑚𝑥 sin 𝑛𝑥 𝑑𝑥 = {0, 𝑚 ̸= 𝑛,𝜋, 𝑚 = 𝑛 ̸= 0.

(37)

It is known that the integral of a periodic function isthe same over any interval whose length equals its period.Therefore, the formulas are valid not only for the interval[−𝜋, 𝜋] but also for any interval [𝑎, 𝑎+2𝜋]; that is, the system(36) is orthogonal on every such interval, where 𝑎 ∈ R.

4. Fourier Series for Fuzzy-ValuedFunctions of Period 2𝜋

Definition 29. Let 𝑓𝑡 be a 2𝜋-periodic fuzzy-valued functionon a set 𝐴. The Fourier series of fuzzy-valued function 𝑓𝑡 ofperiod 2𝜋 is defined as follows:

𝑓𝑡(𝑥) ≅ 𝑎0

⊕⊕

∞

∑

𝑛=1

(𝑎𝑛cos 𝑛𝑥 ⊕ 𝑏

𝑛sin 𝑛𝑥) (38)

with respect to the fuzzy coefficients 𝑎𝑛and 𝑏

𝑛, which

converges uniformly in 𝜆 ∈ [0, 1] for all 𝑛 ∈ N and 𝑥, 𝑡 ∈ 𝐴.


Now, we can calculate the Fourier coefficients 𝑎0, 𝑎𝑛, and

𝑏𝑛with respect to the level sets; that is, 𝑎

𝑛= [(𝑎𝑛)−

𝜆, (𝑎𝑛)+

𝜆]. We

derive from (38) by FH-integrating over [−𝜋, 𝜋] that

[(FH) ∫𝜋

−𝜋


𝜆

= [(FH) ∫𝜋

−𝜋

𝑎0𝑑𝑥]

𝜆

⊕⊕

∞

∑

𝑛=1

[(FH) ∫𝜋

−𝜋

(𝑎𝑛cos 𝑛𝑥 ⊕ 𝑏

𝑛sin 𝑛𝑥) 𝑑𝑥]

𝜆

.

(39)

As an extension of the relation (39) to write with level sets, wehave

[∫

𝜋

−𝜋

𝑓−


𝜋

−𝜋

𝑓+


= [∫

𝜋

−𝜋

(𝑎0)−


𝜋

−𝜋

(𝑎0)+


⊕

∞

∑

𝑛=1

[∫

𝜋

−𝜋

(𝑎𝑛)−

𝜆(𝑡) cos 𝑛𝑡 𝑑𝑡, ∫

𝜋

−𝜋

(𝑎𝑛)+

𝜆(𝑡) cos 𝑛𝑡 𝑑𝑡]

⊕

∞

∑

𝑛=1

[∫

𝜋

−𝜋

(𝑏𝑛)−

𝜆(𝑡) sin 𝑛𝑡 𝑑𝑡, ∫

𝜋

−𝜋

(𝑏𝑛)+

𝜆(𝑡) sin 𝑛𝑡 𝑑𝑡]

(40)

for each 𝜆 ∈ [0, 1] and 𝑥, 𝑡 ∈ [𝑎, 𝑏]. By taking into account theformulas of orthogonal system in (36) for each𝑚, 𝑛 ∈ N with𝑚 ̸= 𝑛, to get 𝑎

𝑛, and by multiplying (39) by cos𝑚𝑥, we obtain

by FH-integrating it over [−𝜋, 𝜋] that

[(FH) ∫𝜋

−𝜋


𝜆

= [(FH) ∫𝜋

−𝜋

𝑎0𝑑𝑥]

𝜆

⊕⊕

∞

∑

𝑛=1

[(FH) ∫𝜋

−𝜋

{𝑎𝑛cos𝑚𝑥 cos 𝑛𝑥

⊕ 𝑏𝑛sin 𝑛𝑥 cos𝑚𝑥} 𝑑𝑥]

𝜆

.

(41)

Similarly to get 𝑏𝑛, multiplying (39) by sin𝑚𝑥 and we

present by FH-integrating it over [−𝜋, 𝜋] that the coefficients𝑎0, 𝑎𝑛, and 𝑏

𝑛with respect to the level sets are derived that

𝑎𝑛=

1

𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥) cos 𝑛𝑥 𝑑𝑥]

𝜆

=

1

𝜋

[∫

𝜋

−𝜋

𝑓−

𝜆(𝑡) cos 𝑛𝑡 𝑑𝑡, ∫

𝜋

−𝜋

𝑓+

𝜆(𝑡) cos 𝑛𝑡 𝑑𝑡] ,

(𝑛 ≥ 0) ,

𝑏𝑛=

1

𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥) sin 𝑛𝑥 𝑑𝑥]

𝜆

=

1

𝜋

[∫

𝜋

−𝜋

𝑓−

𝜆(𝑡) sin 𝑛𝑡 𝑑𝑡, ∫

𝜋

−𝜋

𝑓+

𝜆(𝑡) sin 𝑛𝑡 𝑑𝑡] ,

(𝑛 ≥ 1) .

(42)

Combining the trigonometric identity cos(𝑎 − 𝑏) = cos 𝑎cos 𝑏 + sin 𝑎 sin 𝑏 with 𝑎 = 𝑛𝑠 and 𝑏 = 𝑛𝑥 and substitutingthe formulas (42) in (38), one can observe that

𝑓𝑡(𝑥) ≅

1

2𝜋

[(FH) ∫𝜋

−𝜋


𝜆

⊕⊕

∞

∑

𝑛=1

1

𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥) cos(𝑛𝑠 − 𝑛𝑥)𝑑𝑥]

𝜆

(43)

which is the desired alternate form of the Fourier series offuzzy-valued function 𝑓𝑡 on the interval [−𝜋, 𝜋] for each 𝜆 ∈[0, 1].

Therefore, in looking for a trigonometric series of fuzzy-valued functions whose level sum is a given fuzzy-valuedfunction 𝑓𝑡, it is natural to examine first the series whosecoefficients are given by (42). The trigonometric series withthese coefficients is called the Fourier series of fuzzy-valuedfunction 𝑓𝑡. Incidentally, we note that fuzzy coefficientsinvolve FH-integrating of a fuzzy-valued function of period2𝜋. Therefore, the interval of integration can be replaced byany other interval of length 2𝜋.

Remark 30. Let𝑓𝑡 be any fuzzy-valued function defined onlyon [−𝜋, 𝜋] in trigonometric series. In this case, nothing atall is said about the periodicity of 𝑓𝑡. In fact, if the Fourierseries of fuzzy-valued functions turns out to converge to 𝑓𝑡,then, since it is a periodic function, the level sum of thisautomatically gives us the required periodic extension of 𝑓𝑡.

Example 31. Let 𝑓𝑡 be 2𝜋-periodic fuzzy valued function andFH-integrable on [−𝜋, 𝜋] with trapezoidal form defined by

𝑓𝑡(𝑥) :=

{{{{{{

{{{{{{

{

𝑥 + 𝜋

𝑡 + 𝜋

, −𝜋 ≤ 𝑥 ≤ 𝑡,

1, 𝑡 ≤ 𝑥 ≤ 𝜋 − 𝑡,

𝜋 − 𝑥

𝑡

𝜋 − 𝑡 ≤ 𝑥 ≤ 𝜋,

0, 𝑥 < −𝜋, 𝑥 > 𝜋

(44)

which is FH-integrable on [−𝜋, 𝜋] for each 𝑥, 𝑡 ∈ [𝑎, 𝑏] and𝜆 ∈ [0, 1]. By using Definition 1, the level set [𝑓𝑡]

𝜆of the

membership function 𝑓𝑡 can be written as follows:

[𝑓𝑡]𝜆:= [𝑓−

𝜆(𝑡) , 𝑓+

𝜆(𝑡)] = [𝑡𝜆 + 𝜋 (𝜆 − 1) , 𝜋 − 𝑡𝜆] . (45)

Therefore, we calculate the fuzzy Fourier coefficients 𝑎0, 𝑎𝑛,

and 𝑏𝑛as follows:

𝑎0=

1

2𝜋

[∫

𝜋

−𝜋

[𝑡𝜆 + 𝜋 (𝜆 − 1)] 𝑑𝑡, ∫

𝜋

−𝜋

[𝜋 − 𝑡𝜆] 𝑑𝑡]

= [𝜋 (𝜆 − 1) , 𝜋] ,

𝑎𝑛=

1

𝜋

[∫

𝜋

−𝜋

[𝑡𝜆 + 𝜋 (𝜆 − 1)] cos 𝑛𝑡 𝑑𝑡, ∫𝜋

−𝜋

[𝜋 − 𝑡𝜆] cos 𝑛𝑡 𝑑𝑡]

= [0, 0] = [0]𝜆,


𝑏𝑛=

1

𝜋

[∫

𝜋

−𝜋

[𝑡𝜆 + 𝜋 (𝜆 − 1)] sin 𝑛𝑡 𝑑𝑡, ∫𝜋

−𝜋

[𝜋 − 𝑡𝜆] sin 𝑛𝑡 𝑑𝑡]

= (−1)𝑛[−

2𝜆

𝑛

,

2𝜆

𝑛

] .

(46)

By considering above coefficients in (38) and the condition𝑘[𝑢−

𝜆, 𝑢+

𝜆] = [𝑘𝑢

+

𝜆, 𝑘𝑢−

𝜆] if 𝑘 < 0, we have

𝑓𝑡(𝑥) ≅ [𝜋 (𝜆 − 1) , 𝜋]

⊕⊕

∞

∑

𝑛=1

(−1)𝑛[

−2𝜆

𝑛

,

2𝜆

𝑛

] sin 𝑛𝑥

= [𝜋 (𝜆 − 1) + 2𝜆 sin𝑥 − 𝜆 sin 2𝑥

+

2𝜆

3

sin 3𝑥 − ⋅ ⋅ ⋅ , 𝜋 − 2𝜆 sin𝑥

+ 𝜆 sin 2𝑥 − 2𝜆3

sin 3𝑥 + ⋅ ⋅ ⋅ ] .

(47)

Definition 32 (complex form). Let 𝑓𝑡 be a fuzzy-valued func-tion and FH-integrable on [−𝜋, 𝜋], and its Fourier series is inthe form (38). By substituting Euler’s well-known formulasrelated to the trigonometric and exponential functions: 𝑒𝑖𝑥 =cos𝑥 + 𝑖 sin𝑥 and cos 𝑛𝑥 = (𝑒𝑖𝑛𝑥 + 𝑒−𝑖𝑛𝑥)/2, sin 𝑛𝑥 = (𝑒𝑖𝑛𝑥 −𝑒−𝑖𝑛𝑥

)/2𝑖 in (38), the complex form of Fourier series of fuzzy-valued function 𝑓𝑡 is given by

𝑓𝑡(𝑥) ≅

1

2

𝑎0⊕⊕

∞

∑

𝑛=1

[

1

2

(𝑎𝑛⊕ 𝑖𝑏𝑛) 𝑒𝑖𝑛𝑥⊕

1

2

(𝑎𝑛⊖ 𝑖𝑏𝑛) 𝑒−𝑖𝑛𝑥

] ,

(48)

where the H-difference (𝑎𝑛⊖𝑖𝑏𝑛) exists for all 𝑛 ∈ N and 𝑥, 𝑡 ∈

𝐴.If we set

𝑐0=

1

2

𝑎0, 𝑐

𝑛=

1

2

(𝑎𝑛⊕ 𝑖𝑏𝑛) ,

𝑐−𝑛=

1

2

(𝑎𝑛⊖ 𝑖𝑏𝑛) ,

(49)

and then the𝑀th partial sum of the series (48) and hence ofthe series (38), can be written in the form

𝑠𝑡

𝑀(𝑥) = 𝑐

0⊕⊕

𝑀

∑

𝑛=1

(𝑐𝑛𝑒𝑖𝑛𝑥⊕ 𝑐−𝑛𝑒−𝑖𝑛𝑥

) (50)

Therefore, it is natural to write

𝑓𝑡(𝑥) ≅

⊕

∞

∑

𝑛=−∞

𝑐𝑛𝑒𝑖𝑛𝑥 (51)

The coefficients 𝑐𝑛are given by (49) called the complex

Fourier fuzzy coefficients and satisfy the following relation:

𝑐𝑛=

1

2𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥) 𝑒−𝑖𝑛𝑥

𝑑𝑥]

𝜆

. (52)

Definition 33. Let 𝑓𝑡 be any fuzzy-valued function on [𝑎, 𝑏],defined either on thewhole𝑥-axis or on some intervals.Then,𝑓𝑡 is said to be an even function if 𝑓𝑡(−𝑥) = 𝑓(𝑥) for every𝑥. Thus, the conditions 𝑓−

𝜆(−𝑡) = 𝑓

−


𝜆(−𝑡) = 𝑓

+

𝜆(𝑡)

hold for all 𝑡 ∈ [𝑎, 𝑏] and 𝜆 ∈ [0, 1].

Definition 34. Let 𝑓𝑡 be an even function on [−𝜋, 𝜋], orelse an even periodic function. Then, the Fourier fuzzycoefficients of 𝑓𝑡 are

𝑎𝑛=

1

𝜋

[(FH) ∫𝜋

−𝜋


𝜆

=

2

𝜋

[(FH) ∫𝜋

0


𝜆

(53)

and 𝑏𝑛= [0]𝜆. Therefore, Fourier series of an 𝑓𝑡 consists of

cosines; that is,

𝑓𝑡(𝑥) ≅ 𝑎0

⊕⊕

∞

∑

𝑛=1

𝑎𝑛cos 𝑛𝑥. (54)

Remark 35. By taking into account Definition 13, one canconclude that a fuzzy valued function can not be odd. Becausethe functions 𝑓− and 𝑓+ that are given in RepresentationTheorem can not be odd functions. Therefore, the Fourierseries of fuzzy valued function do not consist of the sines.However, we can define the sines without using the oddnessproperty as follows.

Definition 36. Let 𝑓𝑡 be a periodic fuzzy-valued function onan closed interval. Then, if the fuzzy Fourier coefficient 𝑎

𝑛=

0, then fuzzy Fourier series consists of sines, that is,

𝑓𝑡(𝑥) ≅

⊕

∞

∑

𝑛=1

𝑏𝑛sin 𝑛𝑥 (55)

Definition 37 (one-sided H-derivatives). Let 𝑓𝑡 be any fuzzy-valued function on 𝐴 and continuous except possibly for afinite number of finite jumps.This means that𝑓𝑡 is permittedto be discontinuous at a finite number of points in eachperiod, but at these points we assume that both of theone-sided limits exist and are finite. For convenience, weintroduce this notation for these limits,

𝑓𝑡(𝑥0−) = [ lim

𝑡→ 𝑡0−0

𝑓−

𝜆(𝑡) , lim𝑡→ 𝑡0−0

𝑓+

𝜆(𝑡)]

= [𝑓−

𝜆(𝑡0−) , 𝑓+

𝜆(𝑡0−)] = lim

𝑥→𝑥0−0

𝑓𝑡(𝑥) ,

𝑓𝑡(𝑥0+) = [ lim

𝑡→ 𝑡0+0

𝑓−

𝜆(𝑡) , lim𝑡→ 𝑡0+0

𝑓+

𝜆(𝑡)]

= [𝑓−

𝜆(𝑡0+) , 𝑓+

𝜆(𝑡0+)] = lim

𝑥→𝑥0+0

𝑓𝑡(𝑥)

(56)


for all 𝑥, 𝑡 ∈ 𝐴. In addition, we suppose that the generalizedleft-hand H-derivative (𝑓𝑡

𝐿)

(𝑥0) exists and is defined by

(𝑓𝑡

𝐿)

(𝑥0) = limℎ→0

[

𝑓𝑡(𝑥0+ ℎ) ⊖ 𝑓

𝑡(𝑥0−)

ℎ

]

𝜆

= lim𝑢→0

[

𝑓𝑡(𝑥0− 𝑢) ⊖ 𝑓

𝑡(𝑥0−)

−𝑢

]

𝜆

.

(57)

Thus, we can write

(𝑓𝑡

𝐿)

(𝑥0) = [ limℎ→0

−

𝑓−

𝜆(𝑡0+ ℎ) − 𝑓

−

𝜆(𝑡0−)

ℎ

,

limℎ→0

−

𝑓+

𝜆(𝑡0+ ℎ) − 𝑓

+

𝜆(𝑡0−)

ℎ

]

= [(𝑓−

𝜆)

(𝑡0) , (𝑓+

𝜆)

(𝑡0)] .

(58)

If 𝑓𝑡 is continuous at 𝑥0, this coincides with the usual left-

hand derivative; if 𝑓𝑡 has a discontinuity at 𝑥0, we take care

to use the left-hand instead of just writing 𝑓𝑡(𝑥0).

Symmetrically, we shall also assume that the generalizedright-hand H-derivative (𝑓𝑡

𝑅)

(𝑥0) exists and is defined by

(𝑓𝑡

𝑅)

(𝑥0) = limℎ→0

[

𝑓𝑡(𝑥0+ ℎ) ⊖ 𝑓

𝑡(𝑥0+)

ℎ

]

𝜆

= [ limℎ→0

+

𝑓−

𝜆(𝑡0+ ℎ) − 𝑓

−

𝜆(𝑡0+)

ℎ

,

limℎ→0

+

𝑓+

𝜆(𝑡0+ ℎ) − 𝑓

+

𝜆(𝑡0+)

ℎ

] .

(59)

We begin with quoting the following lemmas which areneeded in proving the convergence of a Fourier series offuzzy-valued functions at each point of discontinuity.

Lemma 38 (see [20, Lemma 2.11.3] (Dirichlet kernel)). TheDirichlet kernel𝐷

𝑁is defined by

𝐷𝑁(𝑢) =

1

2

+

𝑁

∑

𝑛=1

cos 𝑛𝑢, (60)

where 𝑛 is a positive integer. The Dirichlet kernel 𝐷𝑁has the

following two properties. The first involves the definite integralof𝐷𝑁(𝑢) on the interval [0, 𝜋]. That is,

∫

𝜋

0

𝐷𝑁(𝑢) 𝑑𝑢 = ∫

𝜋

0

[

1

2

+

𝑁

∑

𝑛=1

cos 𝑛𝑢]𝑑𝑢 = 𝜋2

(61)

and the second property is

𝐷𝑁(𝑢) =

sin ((2𝑁 + 1) 𝑢/2)2 sin (𝑢/2)

. (62)

Lemma 39. Let 𝑔𝑡 ∈ 𝐶𝐹[0, 𝜋] and FH-integrable on [0, 𝜋[;

then

lim𝑛→∞

[(FH) ∫𝜋

0

𝑔𝑡(𝑢) sin(𝑛𝑢 + 𝑢

2

) 𝑑𝑢]

𝜆

= [0]𝜆, (63)

where 𝑛 is a positive integer.

Proof. By taking into account FH-integration and the Dirich-let kernel defined in Lemma 38, the integral in (63) can beevaluated as

[∫

𝜋

0

𝑔−

𝜆(𝑡) [sin( 𝑡

2

) cos 𝑛𝑡 + cos( 𝑡2

) sin 𝑛𝑡] 𝑑𝑡,

∫

𝜋

0

𝑔+

𝜆(𝑡) [sin( 𝑡

2

) cos 𝑛𝑡 + cos( 𝑡2

) sin 𝑛𝑡] 𝑑𝑡]

=

𝜋

2

[(𝑎𝑛)−

𝜆+ (𝑏𝑛)−

𝜆, (𝑎𝑛)+

𝜆+ (𝑏𝑛)+

𝜆]

=

𝜋

2

𝑎𝑛⊕

𝜋

2

𝑏𝑛,

(64)

where (𝑎𝑛)−

𝜆and (𝑎

𝑛)+

𝜆are the Fourier cosine coefficients

of 𝑔−𝜆(𝑡) sin(𝑡/2) and 𝑔+

𝜆(𝑡) sin(𝑡/2) on the interval ]0, 𝜋[ in

Definition 34. Similarly , (𝑏𝑛)−

𝜆and (𝑏

𝑛)+

𝜆are the Fourier

sine coefficients of 𝑔−𝜆(𝑡) cos(𝑡/2) and 𝑔+

𝜆(𝑡) cos(𝑡/2) on the

interval ]0, 𝜋[ in Definition 36, respectively. Taking the limiton both sides and using orthogonal formulas, we havelim𝑛→∞

𝑎𝑛= 0 and lim

𝑛→∞𝑏𝑛= 0; then we have

lim𝑛→∞

[(FH) ∫𝜋

0

𝑔𝑡(𝑢) sin(𝑛𝑢 + 𝑢

2

) 𝑑𝑢]

𝜆

= lim𝑛→∞

(

𝜋

2

𝑎𝑛⊕

𝜋

2

𝑏𝑛) = [0]𝜆

(65)

for all 𝑢, 𝑡 ∈ [0, 𝜋].

Lemma 40. Suppose that 𝑔𝑡 ∈ 𝐶𝐹[0, 𝜋] and (𝑔𝑡

𝑅)

(0) exists.Then,

lim𝑁→∞

[(FH) ∫𝜋

0

𝑔𝑡(𝑢)𝐷𝑁(𝑢)𝑑𝑢]

𝜆

=

𝜋

2

𝑔𝑡(0+) . (66)

Proof. Let 𝑔𝑡 ∈ 𝐶𝐹[0, 𝜋] and let (𝑔𝑡

𝑅)

(0) exist. Then, we havefrom (66) that

[(FH) ∫𝜋

0

𝑔𝑡(𝑢)𝐷𝑁 (

𝑢) 𝑑𝑢]

𝜆

= [∫

𝜋

0

[𝑔−

𝜆(𝑡) − 𝑔

−

𝜆(0+) + 𝑔

−

𝜆(0+)]𝐷𝑁 (

𝑡) 𝑑𝑡,

∫

𝜋

0

[𝑔+

𝜆(𝑡) − 𝑔

+

𝜆(0+) + 𝑔

+

𝜆(0+)]𝐷

𝑁(𝑡) 𝑑𝑡]

(67)

and this equality turns out to be

[∫

𝜋

0

[𝑔−

𝜆(𝑡) − 𝑔

−

𝜆(0+)]𝐷

𝑁(𝑡) 𝑑𝑡

+ ∫

𝜋

0

𝑔−

𝜆(0+)𝐷

𝑁(𝑡) 𝑑𝑡,

∫

𝜋

0

[𝑔+

𝜆(𝑡) − 𝑔

+

𝜆(0+)]𝐷𝑁 (

𝑡) 𝑑𝑡

+∫

𝜋

0

𝑔+

𝜆(0+)𝐷


(68)


for all 𝑡 ∈ [0, 𝜋] and 𝜆 ∈ [0, 1]. Each of the integrals on theright-hand side will be considered individually. First, usingthe second property of the Dirichlet kernel in (62), we get

∫

𝜋

0

[𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)]𝐷

𝑁(𝑡) 𝑑𝑡

= ∫

𝜋

0

[𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)]

sin (𝑛𝑡 + (𝑡/2))2 sin (𝑡/2)

𝑑𝑡

= ∫

𝜋

0

𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)

2 (𝑡/2)

𝑡/2

sin (𝑡/2)sin(𝑛𝑡 + 𝑡

2

) 𝑑𝑡

= ∫

𝜋

0

𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)

𝑡 − 0

𝑡/2

sin (𝑡/2)sin(𝑛𝑡 + 𝑡

2

) 𝑑𝑡.

(69)

Let ℎ𝑡 be a fuzzy-valued function defined by ℎ𝑡(𝑢) = [𝑔±𝜆(𝑡) −

𝑔±

𝜆(0+)]𝑡/[2(𝑡 − 0) sin(𝑡/2)] and continuous on ]0, 𝜋]. For the

sake of argument, itmust be established thatℎ±𝜆(𝑡) is piecewise

continuous on (0, 𝜋).The piecewise continuity of ℎ±𝜆(𝑡) hinges

on the right-side limit at 𝑡 = 0.Consider

lim𝑡→0+

ℎ𝑡(𝑢) = lim

𝑡→0+

𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)

𝑡 − 0

𝑡/2

sin (𝑡/2). (70)

Provided that the individual limits at (68) exist. The continu-ity of ℎ𝑡 allows the application of Lemma 39, so that

lim𝑁→∞

∫

𝜋

0

[𝑔±

𝜆(𝑡) − 𝑔

±

𝜆(0+)]𝐷𝑁 (

𝑡) 𝑑𝑡

= lim𝑁→∞

[ℎ𝑡(𝑢) sin(𝑛𝑢 + 𝑢

2

) 𝑑𝑢]

𝜆

= [0]𝜆.

(71)

As for the second integral on (68), it follows that

lim𝑁→∞

∫

𝜋

0

𝑔±

𝜆(0+)𝐷𝑁 (

𝑡) 𝑑𝑡 =

𝜋

2

𝑔±

𝜆(0+) . (72)

Combining the results, it follows that

lim𝑁→∞

[(FH) ∫𝜋

0

𝑔𝑡(𝑢)𝐷𝑁(𝑢)𝑑𝑢]

𝜆

= [ lim𝑁→∞

∫

𝜋

0

[𝑔−

𝜆(𝑡) − 𝑔

−

𝜆(0+)]𝐷

𝑁(𝑡) 𝑑𝑡

+ lim𝑁→∞

∫

𝜋

0

𝑔−

𝜆(0+)𝐷

𝑁(𝑡) 𝑑𝑡,

lim𝑁→∞

∫

𝜋

0

[𝑔+

𝜆(𝑡) − 𝑔

+

𝜆(0+)]𝐷

𝑁(𝑡) 𝑑𝑡

+ lim𝑁→∞

∫

𝜋

0

𝑔+

𝜆(0+)𝐷


= [

𝜋

2

𝑔−

𝜆(0+) ,

𝜋

2

𝑔+

𝜆(0+)] =

𝜋

2

𝑔𝑡(0+) .

(73)

Theorem 41. Let 𝑓𝑡 be any 2𝜋-periodic continuous fuzzy-valued function and H-differentiable on [−𝜋, 𝜋]. The Fourierseries of fuzzy-valued function converges to

(i) 𝑓𝑡(𝑥) for every value 𝑥, where 𝑓𝑡 ∈ 𝐶𝐹[−𝜋, 𝜋] for each

𝜆 ∈ [0, 1],(ii) the arithmetic mean of the right-hand and left-hand

limits 𝑓𝑡(𝑥−) and 𝑓𝑡(𝑥+) which are given in Defini-tion 37, where the one-sided limits at each point ofdiscontinuity exist.

Proof. (i) Firstly, continuity and the existence of one-sidedH-derivatives are sufficient for convergence. Secondly, if 𝑓𝑡 ∈𝐶𝐹[−𝜋, 𝜋] at 𝑥, it follows that 𝑓𝑡(𝑥+) = 𝑓𝑡(𝑥) = 𝑓𝑡(𝑥−), so

the Fourier series of fuzzy-valued function converges to𝑓𝑡(𝑥)for all 𝑥, 𝑡 ∈ [−𝜋, 𝜋] and for each 𝜆 ∈ [0, 1].

(ii) The continuity means that Fourier fuzzy coefficients𝑎𝑛and 𝑏

𝑛exist for all appropriate values of 𝑛, and the

corresponding Fourier series for 𝑓𝑡 is given by (43). The𝑁thpartial level sum 𝑆

𝑁of the series in (43) is

𝑓𝑡(𝑥) ≅

1

2𝜋

[(FH) ∫𝜋

−𝜋


𝜆

⊕⊕

𝑁

∑

𝑛=1

1

𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥) cos (𝑛𝑠 − 𝑛𝑥) 𝑑𝑥]

𝜆

.

(74)

Since the first property of Dirichlet kernel𝐷𝑁(𝑠−𝑥) = (1/2)+

∑𝑁

𝑛=1cos(𝑛𝑠 − 𝑛𝑥), using the partial level sum in (74), we get

𝑆𝑡

𝑁(𝑥) =

1

𝜋

[(FH) ∫𝜋

−𝜋

𝑓𝑡(𝑥)𝐷𝑁 (

𝑠 − 𝑥) 𝑑𝑥]

𝜆

=

1

𝜋

[∫

𝜋

−𝜋

𝑓−

𝜆(𝑡) 𝐷𝑁 (

𝑠 − 𝑡) 𝑑𝑡,

∫

𝜋

−𝜋

𝑓+

𝜆(𝑡) 𝐷𝑁(𝑠 − 𝑡) 𝑑𝑡]

(75)

for 𝑥, 𝑡 ∈ [−𝜋, 𝜋] and 𝑠 ∈ R. By using 2𝜋-periodicity of 𝑓𝑡and the Dirichlet kernel in Lemma 38, we have

𝑆𝑡

𝑁(𝑥) =

1

𝜋

[∫

𝑡+𝜋

𝑡−𝜋

𝑓−



∫

𝑡+𝜋

𝑡−𝜋

𝑓+


𝑠 − 𝑡) 𝑑𝑡] .

(76)

The integral in (76) splits into the two following integrals:

𝑆𝑡

𝑁(𝑥) =

1

𝜋

∫

𝑡

𝑡−𝜋

[𝑓−



𝑓+


+

1

𝜋

∫

𝑡+𝜋

𝑡

[𝑓−

𝜆(𝑡) 𝐷𝑁(𝑠 − 𝑡) 𝑑𝑡,

𝑓+

𝜆(𝑡) 𝐷𝑁(𝑠 − 𝑡) 𝑑𝑡] .

(77)

Each integral on the right-hand side can be simplified usingLemma 40, after making an appropriate change of variable.


For the first integral, the change of variable will be 𝑢 = −𝑡 + 𝑠so that

∫

𝑡

𝑡−𝜋

𝑓±

𝜆(𝑡) 𝐷𝑁(𝑠 − 𝑡) 𝑑𝑡

= −∫

0

𝜋

𝑓±

𝜆(𝑠 − 𝑢)𝐷

𝑁(𝑢) 𝑑𝑢

= ∫

𝜋

0

𝑓±

𝜆(𝑠 − 𝑢)𝐷𝑁 (

𝑢) 𝑑𝑢.

(78)

Suppose that [𝑓−𝜆(𝑠−𝑢), 𝑓

+

𝜆(𝑠−𝑢)] = [𝑓

𝑠−𝑢(𝑡0)]𝜆= [𝑔𝑢(𝑡0)]𝜆=

[𝑔−

𝜆(𝑢), 𝑔+

𝜆(𝑢)] for all 𝑡

0∈ [0, 𝜋] in (78). Since the functions

𝑔±

𝜆are piecewise continuous on ]0, 𝜋[ and 𝑔𝑢

𝑅(0) exists, to

establish the existence of the right-hand H-derivative of𝑔𝑢(𝑡0) at 𝑡0= 0, we have

(𝑔𝑢)

𝑅(0) = lim

𝑡→0+

[

𝑔𝑢(𝑡) ⊖ 𝑔

𝑢(0+)

𝑡 ⊖ 0+

]

𝜆

, (79)

where 𝑔𝑢(0+) = lim𝑡→0+

[𝑔𝑢(𝑡)]𝜆= lim

𝑡→0+[𝑓𝑠−𝑢(𝑡)]𝜆=

lim𝑠→𝑢

𝑓𝑡(𝑠 − 𝑢) = 𝑓

𝑡(𝑥−). Consequently, we derive that

lim𝑁→∞

∫

𝑡

𝑡−𝜋

𝑓±


𝑠 − 𝑡) 𝑑𝑡

= lim𝑁→∞

∫

𝜋

0

𝑓±

𝜆(𝑠 − 𝑢)𝐷

𝑁(𝑢) 𝑑𝑢

= lim𝑁→∞

∫

𝜋

0

𝑔±

𝜆(𝑢)𝐷𝑁(𝑢) 𝑑𝑢

=

𝜋

2

𝑔𝑢(0+) =

𝜋

2

𝑓𝑡(𝑥−) .

(80)

The second integral on the right-hand side of (77) is analysedin a similar way. In this case, the change of variable is 𝑢 = 𝑡−𝑠.Suppose that if we take [𝑓−

𝜆(𝑠+𝑢), 𝑓

+

𝜆(𝑠+𝑢)] = [𝑔

−

𝜆(𝑢), 𝑔+

𝜆(𝑢)],

then

lim𝑁→∞

∫

𝑡+𝜋

𝑡

𝑓±

𝜆(𝑡) 𝐷𝑁(𝑡 − 𝑠) 𝑑𝑡

= lim𝑁→∞

∫

𝜋

0

𝑔±

𝜆(𝑢)𝐷𝑁(𝑢) 𝑑𝑢

=

𝜋

2

𝑔𝑢(0+) =

𝜋

2

𝑓𝑡(𝑥+) .

(81)

By taking into account (80) and (81), and if we let 𝑁 →∞ in (77), then we have

lim𝑁→∞

1

𝜋

[∫

𝑡

𝑡−𝜋

[𝑓−


𝑓+


+ ∫

𝑡+𝜋

𝑡

[𝑓−


𝑓+

𝜆(𝑡) 𝐷𝑁(𝑠 − 𝑡) 𝑑𝑡] ]

=

1

2

(𝑓𝑡(𝑥−) + 𝑓

𝑡(𝑥+)) .

(82)

This completes the proof.

We assume that the above results hold with respect to2𝜋-periodic fuzzy-valued functions. The similar results canbe obtained for a continuous H-differentiable periodic fuzzy-valued function of an arbitrary period 𝑃 > 0.

5. Conclusion

As conventional hardware systems have been based onmem-bership functions, a membership grade has been assignedto each element in the universe of discourse [21]. In thisway, a wide variety of membership-function forms are beingimplemented and may reduce the number of conditionalpropositions for fuzzy inference to generate complex nonlin-ear surfaces, such as those used in fuzzy control and fuzzymodeling. More complex surfaces can be generated with alimited number of conditional propositions, with increasingtypes of membership-function forms. This is an advantageover approximating membership functions, especially withtriangular or trapezoidal forms. Indeed, some useful resultshave been obtained by using level sets for defining seriesof fuzzy-valued functions like Fourier series. The potentialapplications of the obtained results include the generalizationof sequences and series of fuzzy-valued functions.

One of the purposes of this work is to extend the classicalanalysis to the fuzzy level set analysis dealing with fuzzy-valued functions. Some of the analogies are demonstrated bytheoretical examples between classical and level set calculus.Of course, several possible applications on Fourier series overreal or complex field can be extended to the fuzzy numberspace. We should record from now on that the main resultsgiven in Section 4 of the present paper will be based onexamining Fourier analysis of fuzzy-valued functions. Futurework will be dedicated to find some applications on Fourierseries of these functions.

Conflict of Interests

The authors declare that there is no conflict of interests re-garding the publication of this paper.

References

[1] M. Stojaković and Z. Stojaković, “Series of fuzzy sets,” Fuzzy Setsand Systems, vol. 160, no. 21, pp. 3115–3127, 2009.

[2] M. Stojaković and Z. Stojaković, “Addition and series of fuzzysets,” Fuzzy Sets and Systems, vol. 83, no. 3, pp. 341–346, 1996.

[3] Ö. Talo and F. Başar, “On the space 𝑏V𝑝(𝐹) of sequences of p-

bounded variation of fuzzy numbers,”ActaMathematica Sinica,vol. 24, no. 6, pp. 965–972, 2008.

[4] U. Kadak and F. Başar, “Power series of fuzzy numbers,” inProceedings of the International Conference on MathematicalScience (ICMS ’10), pp. 538–550, November 2010.

[5] U. Kadak and F. Başar, “Power series of fuzzy numbers with reelor fuzzy coefficients,” Filomat, vol. 25, no. 3, pp. 519–528, 2012.

[6] U. Kadak and F. Başar, “On some sets of fuzzy-valued sequenceswith the level sets,” Contemporary Analysis and Applied Mathe-matics, vol. 1, no. 2, pp. 70–90, 2013.

[7] L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3,pp. 338–353, 1965.


[8] R. Goetschel Jr. and W. Voxman, “Elementary fuzzy calculus,”Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986.

[9] K. H. Lee, First Course on Fuzzy Theory and Applications,Springer, Berlin, Germany, 2005.

[10] P. Diamond and P. Kloeden, “Metric spaces of fuzzy sets,” FuzzySets and Systems, vol. 35, no. 2, pp. 241–249, 1990.

[11] S. Nanda, “On sequences of fuzzy numbers,” Fuzzy Sets andSystems, vol. 33, no. 1, pp. 123–126, 1989.

[12] Ö. Talo and F. Başar, “Determination of the duals of classical setsof sequences of fuzzy numbers and related matrix transforma-tions,” Computers and Mathematics with Applications, vol. 58,no. 4, pp. 717–733, 2009.

[13] B. Bede and S. G. Gal, “Almost periodic fuzzy-number-valuedfunctions,” Fuzzy Sets and Systems, vol. 147, no. 3, pp. 385–403,2004.

[14] M. L. Puri and D. A. Ralescu, “Differentials of fuzzy functions,”Journal of Mathematical Analysis and Applications, vol. 91, no. 2,pp. 552–558, 1983.

[15] L. Stefanini, “A generalization of Hukuhara difference,”Advances in Soft Computing, vol. 48, pp. 203–210, 2008.

[16] L. Stefanini and B. Bede, “Generalized Hukuhara differentiabil-ity of interval-valued functions and interval differential equa-tions,” Nonlinear Analysis, Theory, Methods and Applications,vol. 71, no. 3-4, pp. 1311–1328, 2009.

[17] U. Kadak, “On the sets of fuzzy-valued function with the levelsets,” Journal of Fuzzy Set Valued Analysis, vol. 2013, pp. 1–13,2013.

[18] M. Hukuhara, “Integration des applications mesurables dont lavaleur est un compact convex,” Funkcialaj Ekvacioj, vol. 10, pp.205–229, 1967.

[19] G. A. Anastassiou, “Fuzzymathematics: approximation theory,”Studies in Fuzziness and SoftComputing, vol. 251, pp. 1–451, 2010.

[20] R. Bernatz, Fourier Series and Numerical Methods for PartialDifferential Equations, John Wiley & Sons, Hoboken, NJ, USA,2010.

[21] Y. K. Kim and B. M. Ghil, “Integrals of fuzzy-number-valuedfunctions,” Fuzzy Sets and Systems, vol. 86, pp. 213–222, 1997.

Submit your manuscripts athttp://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

research article on fourier series of fuzzy-valued functions · 2019. 7. 31. · on fourier series...

Documents