On Fourier Series of Fuzzy-Valued Functions

Fourier analysis is a powerful tool for many problems, and especially for solving various differential equations of interest in science and engineering. In the present paper since the utilization of Zadeh's Extension principle is quite difficult in practice, we prefer the idea of level sets in order to construct a fuzzy-valued function on a closed interval via related membership function. We derive uniform convergence of a fuzzy-valued function sequences and series with level sets. Also we study Hukuhara differentiation and Henstock 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 functions at each point of discontinuity, where one-sided limits exist.


Introduction
Fourier series were introduced by Joseph Fourier (1768-1830) for the purpose of solving the heat equation in a metal plate and it has long provided one of the principal methods of analysis for mathematical physics, engineering, and signal processing. While the original theory of Fourier series applies to the periodic functions occurring in wave motion, such as with light and sound, its generalizations often relate to wider settings, such as the time-frequency analysis underlying the recent theories of wavelet analysis and local trigonometric analysis. Additionally, the idea of Fourier was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves and to write the solution as a superposition of the corresponding eigen solutions. This superposition or linear combination is called the Fourier series.
Due to the rapid development of the fuzzy theory, however, some of these basic concepts have been modified and improved. One of them set mapping operations to the case of interval valued fuzzy sets. To accomplish this, we need to introduce the idea of the level sets of interval fuzzy sets and the related formulation of a representation of an interval valued fuzzy set in terms of its level sets. Once having these structures, we can then provide the desired extension to interval valued fuzzy sets. The effectiveness of level sets comes from not only their required memory capacity for fuzzy sets, but also from their two valued nature. This nature contributes to an effective derivation of the fuzzy-inference algorithm based on the families of the level sets. Besides, the definition of fuzzy sets by level sets offers advantages over membership functions, especially when the fuzzy sets are in universes of discourse with many elements.
Furthermore, we also study the Fourier series of periodic fuzzy-valued functions. Using a different approach, it can be shown that the Fourier series with fuzzy coefficients converges. Applying this idea, we establish some connections between the Fourier series and Fourier series of fuzzyvalued functions with the level sets. Quite recently, by using Zadeh's Extension Principle, M. Stojaković and Z. Stojaković investigated the convergence of series of fuzzy numbers in [1] and they gave some results which complete their previous results in [2]. Additionally, Talo and Başar [3] have extended the main results related to the sequence spaces and matrix 2 The Scientific World Journal transformations on the real or complex field to the fuzzy numbers with the level sets. Also, Kadak and Başar [4,5] have recently studied the power series of fuzzy numbers and examined on some sets of fuzzy-valued sequences with the level sets and gave some properties of the level sets together with some inclusion relations in [6].
The rest of this paper is organized as follows. In Section 2, we give some required definitions and consequences related to the fuzzy numbers, sequences, and series of fuzzy numbers. We also report the most relevant and recent literature in this section. In Section 3, first, the definition of periodic fuzzyvalued function is given which will be used in the proof of our main results. In this section, Hukuhara differentiation and Henstock integration are presented according to fuzzyvalued functions which depend on , ∈ [ , ]. This section is terminated with the condensation of the results on uniform convergence of fuzzy-valued sequences and series. In the final section of the paper, we assert that the Fourier series of a fuzzy-valued function with 2 period converges and especially prove the convergence about a discontinuity point by using Dirichlet kernel and one-sided limits.

Preliminaries, Background, and Notation
A fuzzy number is a fuzzy set on the real axis; that is, a mapping : R → [0, 1] which satisfies the following four conditions.
We denote the set of all fuzzy numbers on R by 1 and called it the space of fuzzy numbers. -level set [ ] of ∈ 1 is defined by (2) Representation Theorem (see [8] Definition 1 ((trapezoidal fuzzy number) [9, Definition, p. 145]). We can define trapezoidal fuzzy number as = ( 1 , 2 , 3 , 4 ); the membership function ( ) of this fuzzy number will be interpreted as follows: Then, the result [ ] : Let , V, ∈ 1 and ∈ R. Then the operations addition, scalar multiplication and product defined on 1 by where it is immediate that for all ∈ [0, 1]. Let be the set of all closed bounded intervals of real numbers with endpoints and ; that is, . Define the relation on by Then it can easily be observed that is a metric on (cf. Diamond and Kloeden [10]) and ( , ) is a complete metric The Scientific World Journal 3 space (cf. Nanda [11]). Now, we can define the metric on 1 by means of the Hausdorff metric as Definition 2 (see [12], Definition 2.1).
∈ 1 is said to be a nonnegative fuzzy number if and only if ( ) = 0 for all < 0. It is immediate that ⪰ 0 if is a nonnegative fuzzy number.
One can see that Proposition 3 (see [13]). Let , V, , ∈ 1 and ∈ R. Then, is a complete metric space, (cf. Puri and Ralescu [14]).  Remark 5 (see [12]). According to Definition 4, the following remarks can be given. Definition 6 (see [12]). Let ( ) ∈ ( ). Then the expression ⊕ ∑ is called a series of fuzzy numbers with the level summation ⊕ ∑. Define the sequence ( ) via th partial level sum of the series by for all ∈ N. If the sequence ( ) converges to a fuzzy number then we say that the series ⊕ ∑ of fuzzy numbers converges to and write ⊕ ∑ which implies that where the summation is in the sense of classical summation and converges uniformly in ∈ [0, 1]. Conversely, if satisfy the conditions of Theorem 8.
Theorem 9 (cf. [13]). The following statements for level addition ⊕ of fuzzy numbers and classical addition + of real scalars are valid.
(ii) With respect to 0, none of ̸ = , ∈ R has opposite in 1 .
For general , ∈ R, the above property does not hold.

Generalized Hukuhara Difference.
Let K be the space of nonempty compact and convex sets in the -dimensional Euclidean space R . If = 1, denote by the set of (closed bounded) intervals of the real line. Given two elements , ∈ K and ∈ R, the usual interval arithmetic operations, that is, addition and scalar multiplication, are defined by + = { + : ∈ , ∈ } and = { : ∈ }. It is well known that addition is associative and commutative and with neutral element {0}. If = −1, scalar multiplication gives the opposite −A = (−1) = {− : ∈ } but, in general, + (− ) ̸ = 0; that is, the opposite of is not the inverse of in addition unless is a singleton. A first consequence of this fact is that, in general, additive simplification is not valid.
To partially overcome this situation, the Hukuhara difference, H-difference for short, has been introduced as a set for which ⊖ ⇔ = + and an important property of ⊖ is that ⊖ = {0} for all ∈ K and ( + ) ⊖ = for all , ∈ K. The H-difference is unique, but it does not always exist. A necessary condition for ⊖ to exist is that contains a translation { } + of . A generalization of the Hukuhara difference proposed in [15] aims to overcome this situation.
(a) Let , ∈ K be two compact convex sets. Then, we have that (i) if the H-difference exists, it is unique and is a generalization of the usual Hukuhara difference since ⊖ = − , whenever ⊖ exists.  Proposition 12 (see [16]). The following statements hold.
Then, we have where the limits are in the Hausdorff metric for intervals.

Fuzzy-Valued Functions with the Level Sets
In this chapter, we consider sequences and series of fuzzy-valued function and develop uniform convergence, Hukuhara differentiation, and Henstock integration. In addition, we present characterizations of uniform convergence signs in sequences of fuzzy-valued functions.
Definition 13 (see [6]). Consider a function from [ , ] into 1 with respect to a membership function which is called trapezoidal fuzzy number and is interpreted as follows: Then, the membership function turns out to be ( ) =

Generalized Hukuhara Differentiation.
The concept of fuzzy differentiability comes from a generalization of the Hukuhara difference for compact convex sets. We prove several properties of the derivative of fuzzy-valued functions considered here. As a continuation of Hukuhara derivatives for real fuzzy-valued functions [18], we can define Hdifferentiation of a fuzzy-valued function with respect to level sets. For short, throughout the paper, we write instead of "Hukuhara sense. " (1) ( ) ( ) ∈ 1 exists such that, for all ℎ > 0 sufficiently near to 0, the H-difference ( + ℎ) ⊖ ( ) exists; then the H-derivative ( ) ( ) is given as follows: or (2) ( ) ( ) ∈ 1 exists such that, for all ℎ < 0 sufficiently near to 0, the H-difference ( + ℎ) ⊖ ( ) exists; then the H-derivative ( ) ( ) is given as follows: for all , ∈ [ , ] and ∈ [0, 1].
From here, we remember that the H-derivative of at , ∈ [ , ] depends on the value and the choice of a constant ∈ [0, 1].

Generalized Fuzzy-Henstock Integration
Definition 18 (see [19,Definition 8.7]). A fuzzy valued function is said to be fuzzy-Henstock, in short FH-integrable, if for any > 0, there exists > 0 such that Remark 21. Note that if is periodic fuzzy-valued function and FH-integrable on any interval of length , then it is FHintegrable on any other of the same length, and the value of the integral is the same; that is, for all , ∈ [ , ] and ∈ [0, 1].
This property is an immediate consequence of the interpretation of an integral as an area. In fact, each integral (24) equals the area bounded by the curves ± ( ), the straight lines = and = , and the closed interval [ , ] of -axis. In the present case, the areas represented by two integrals are the same because of the periodicity of . Hereafter, when we say that a fuzzy-valued function with period is FHintegrable, we mean that it is FH-integrable on an interval of length . It follows from the property just proved that is also FH-integrable on any interval of finite length.
In other words, { ( )} converges to on if and only if for each ∈ and for an arbitrary > 0, there exists an integer = ( , ) such that ( ( ), ( )) < whenever > . The integer in the definition of pointwise convergence may, in general, depend on both > 0 and ∈ . If, however, one integer can be found that works for all points in , then the convergence is said to be uniform. That is, a sequence of fuzzy-valued functions { ( )} converges uniformly to on a set if, for each > 0, there exists an integer ( ) such that Theorem 23 (see [6]). Let , ∈ and ∈ [0, 1]. Then, the following statements are valid.

(i) A sequence of fuzzy-valued functions { ( )} defined on a set ⊆ R converges uniformly to a fuzzy-valued function on if and only if
= sup and the equality on rigt-hand side in (32) is evaluated as for > ( ). Since is arbitrary, this step completes the proof.
The hypothesis of Theorem 24 is sufficient for our purposes and may be used to show the nonuniform convergence of the sequence { ( )} on [ , ]. Also, it is important to point out that a direct analogue of Theorem 24 for H-derivatives is not true.

Corollary 28 (interchange of summation and integration). Suppose that { ( )} is a sequence in [ , ] and
( ) converges uniformly to ( ) on [ , ]. Then, where (FH) ∫ ( ) exists for all , ∈ [ , ] and ∈ [0, 1]. Now, we give an important trigonometric system whose special case of one of the systems of functions is applying to the well-known inequalities. By a trigonometric system we mean the system of 2 periodic and functions which is given by 1, cos ( ) , sin ( ) , cos (2 ) , sin (2 ) , . . . , for all ∈ N. We now prove some auxiliary formulas for any positive integer such that ∫ − cos( ) = ∫ − sin( ) = 0. Therefore, one can see by using trigonometric identities that It is known that the integral of a periodic function is the same over any interval whose length equals its period. Therefore, the formulas are valid not only for the interval [− , ] but also for any interval [ , + 2 ]; that is, the system (36) is orthogonal on every such interval, where ∈ R.

Fourier Series for Fuzzy-Valued Functions of Period 2
Definition 29. Let be a 2 -periodic fuzzy-valued function on a set . The Fourier series of fuzzy-valued function of period 2 is defined as follows: with respect to the fuzzy coefficients and , which converges uniformly in ∈ [0, 1] for all ∈ N and , ∈ .
As an extension of the relation (39) to write with level sets, we have Combining the trigonometric identity cos( − ) = cos cos + sin sin with = and = and substituting the formulas (42) in (38), one can observe that which is the desired alternate form of the Fourier series of fuzzy-valued function on the interval [− , ] for each ∈ [0, 1]. Therefore, in looking for a trigonometric series of fuzzyvalued functions whose level sum is a given fuzzy-valued function , it is natural to examine first the series whose coefficients are given by (42). The trigonometric series with these coefficients is called the Fourier series of fuzzy-valued function . Incidentally, we note that fuzzy coefficients involve FH-integrating of a fuzzy-valued function of period 2 . Therefore, the interval of integration can be replaced by any other interval of length 2 .
Remark 30. Let be any fuzzy-valued function defined only on [− , ] in trigonometric series. In this case, nothing at all is said about the periodicity of . In fact, if the Fourier series of fuzzy-valued functions turns out to converge to , then, since it is a periodic function, the level sum of this automatically gives us the required periodic extension of .
By considering above coefficients in (38) and the condition Definition 32 (complex form). Let be a fuzzy-valued function and FH-integrable on [− , ], and its Fourier series is in the form (38). By substituting Euler's well-known formulas related to the trigonometric and exponential functions: = cos + sin and cos = ( (38), the complex form of Fourier series of fuzzyvalued function is given by where the H-difference ( ⊖ ) exists for all ∈ N and , ∈ . If we set and then the th partial sum of the series (48) and hence of the series (38), can be written in the form Therefore, it is natural to write The coefficients are given by (49) called the complex Fourier fuzzy coefficients and satisfy the following relation: . Therefore, Fourier series of an consists of cosines; that is, cos . (54) Remark 35. By taking into account Definition 13, one can conclude that a fuzzy valued function can not be odd. Because the functions − and + that are given in Representation Theorem can not be odd functions. Therefore, the Fourier series of fuzzy valued function do not consist of the sines. However, we can define the sines without using the oddness property as follows.
Definition 36. Let be a periodic fuzzy-valued function on an closed interval. Then, if the fuzzy Fourier coefficient = 0, then fuzzy Fourier series consists of sines, that is, Definition 37 (one-sided H-derivatives). Let be any fuzzyvalued function on and continuous except possibly for a finite number of finite jumps. This means that is permitted to be discontinuous at a finite number of points in each period, but at these points we assume that both of the one-sided limits exist and are finite. For convenience, we introduce this notation for these limits, 10 The Scientific World Journal for all , ∈ . In addition, we suppose that the generalized left-hand H-derivative ( ) ( 0 ) exists and is defined by (57) Thus, we can write If is continuous at 0 , this coincides with the usual lefthand derivative; if has a discontinuity at 0 , we take care to use the left-hand instead of just writing ( 0 ).
Symmetrically, we shall also assume that the generalized right-hand H-derivative ( ) ( 0 ) exists and is defined by We begin with quoting the following lemmas which are needed in proving the convergence of a Fourier series of fuzzy-valued functions at each point of discontinuity.
Lemma 38 (see [20,Lemma 2.11.3] (Dirichlet kernel)). The Dirichlet kernel is defined by where is a positive integer. The Dirichlet kernel has the following two properties. The first involves the definite integral of ( ) on the interval [0, ]. That is, where is a positive integer.
Proof. By taking into account FH-integration and the Dirichlet kernel defined in Lemma 38, the integral in (63) can be evaluated as Proof. Let ∈ [0, ] and let ( ) (0) exist. Then, we have from (66) that and this equality turns out to be The Scientific World Journal 11 for all ∈ [0, ] and ∈ [0, 1]. Each of the integrals on the right-hand side will be considered individually. First, using the second property of the Dirichlet kernel in (62), we get Let ℎ be a fuzzy-valued function defined by ℎ ( ) = [ ± ( ) − ± (0+)] /[2( − 0) sin( /2)] and continuous on ]0, ]. For the sake of argument, it must be established that ℎ ± ( ) is piecewise continuous on (0, ). The piecewise continuity of ℎ ± ( ) hinges on the right-side limit at = 0. Consider Provided that the individual limits at (68) exist. The continuity of ℎ allows the application of Lemma 39, so that As for the second integral on (68), it follows that Combining the results, it follows that  (ii) The continuity means that Fourier fuzzy coefficients and exist for all appropriate values of , and the corresponding Fourier series for is given by (43). The th partial level sum of the series in (43) is Since the first property of Dirichlet kernel ( − ) = (1/2)+ ∑ =1 cos( − ), using the partial level sum in (74), we get for , ∈ [− , ] and ∈ R. By using 2 -periodicity of and the Dirichlet kernel in Lemma 38, we have The integral in (76) splits into the two following integrals: Each integral on the right-hand side can be simplified using Lemma 40, after making an appropriate change of variable.
This completes the proof.
We assume that the above results hold with respect to 2 -periodic fuzzy-valued functions. The similar results can be obtained for a continuous H-differentiable periodic fuzzyvalued function of an arbitrary period > 0.

Conclusion
As conventional hardware systems have been based on membership functions, a membership grade has been assigned to each element in the universe of discourse [21]. In this way, a wide variety of membership-function forms are being implemented and may reduce the number of conditional propositions for fuzzy inference to generate complex nonlinear surfaces, such as those used in fuzzy control and fuzzy modeling. More complex surfaces can be generated with a limited number of conditional propositions, with increasing types of membership-function forms. This is an advantage over approximating membership functions, especially with triangular or trapezoidal forms. Indeed, some useful results have been obtained by using level sets for defining series of fuzzy-valued functions like Fourier series. The potential applications of the obtained results include the generalization of sequences and series of fuzzy-valued functions.
One of the purposes of this work is to extend the classical analysis to the fuzzy level set analysis dealing with fuzzyvalued functions. Some of the analogies are demonstrated by theoretical examples between classical and level set calculus. Of course, several possible applications on Fourier series over real or complex field can be extended to the fuzzy number space. We should record from now on that the main results given in Section 4 of the present paper will be based on examining Fourier analysis of fuzzy-valued functions. Future work will be dedicated to find some applications on Fourier series of these functions.