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.
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 fuzzy-valued 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  and they gave some results which complete their previous results in . Additionally, Talo and Başar  have extended the main results related to the sequence spaces and matrix 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 .
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 fuzzy-valued 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 fuzzy-valued 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 period converges and especially prove the convergence about a discontinuity point by 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 mapping which satisfies the following four conditions.(i) is normal; that is, there exists an such that .(ii) is fuzzy convex; that is, for all and for all .(iii) is upper semicontinuous.(iv)The set is compact (cf. Zadeh ), where denotes the closure of the set in the usual topology of .We denote the set of all fuzzy numbers on by and called it the space of fuzzy numbers. -level set of is defined by The set is closed, bounded and, nonempty interval for each which is defined by . can be embedded in , since each can be regarded as a fuzzy number defined by
Representation Theorem (see ). Let for for each . Then the following statements hold.(i) is a bounded and nondecreasing left continuous function on .(ii) is a bounded and nonincreasing left continuous function on .(iii) The functions and are right continuous at the point .(iv).
Conversely, if the pair of functions and satisfies the conditions (i)–(iv), then there exists a unique such that for each . The fuzzy number corresponding to the pair of functions and is defined by , .
Definition 1 ((trapezoidal fuzzy number) [9, Definition, p. 145]). We can define trapezoidal fuzzy number as ; the membership function of this fuzzy number will be interpreted as follows: Then, the result holds for each .
Let and . Then the operations addition, scalar multiplication and product defined on by where it is immediate that for all . 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 ) and is a complete metric space (cf. Nanda ). Now, we can define the metric on by means of the Hausdorff metric as
Definition 2 (see , Definition 2.1). is said to be a nonnegative fuzzy number if and only if for all . It is immediate that if is a nonnegative fuzzy number.
One can see that
Proposition 3 (see ). Let and . Then, (i) is a complete metric space, (cf. Puri and Ralescu ).(ii).(iii).(iv).(v).
Definition 4. The following basic statements hold. (i)[12, Definition 2.7] A sequence of fuzzy numbers is a function from the set into the set . The fuzzy number denotes the value of the function at and is called as the general term of the sequence. By , we denote the set of all sequences of fuzzy numbers.(ii)[12, Definition 2.9] A sequence is called convergent with limit if and only if for every there exists such that for all .(iii)[12, Definition 2.11] A sequence is called bounded if and only if the set of fuzzy numbers consisting of the terms of the sequence is a bounded set; that is to say, a sequence is bounded if and only if there exist two fuzzy numbers and such that for all . This means that and for all .
Remark 5 (see ). According to Definition 4, the following remarks can be given.(a)Obviously the sequence converges to a fuzzy number if and only if and converge uniformly to and on , respectively.(b)The boundedness of the sequence is equivalent to the fact that If the sequence is bounded then the sequences of functions and are uniformly bounded in .
Definition 6 (see ). Let . Then the expression is called a series of fuzzy numbers with the level summation . Define the sequence via partial level sum of the series by for all . 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 . Conversely, if converge uniformly in , then defines a fuzzy number such that .
Definition 7 (see [12, Definition 2.14]). Let be a sequence of functions defined on and . Then, is said to be eventually equi-left-continuous at if for any there exist and such that whenever and . Similarly, eventually equi-right-continuity at of can be defined.
Theorem 8 (see [12, Theorem 2.15]). Let such that and , as for each . Then, the pair of functions and determines a fuzzy number if and only if the sequences of functions and are eventually equi-left-continuous at each and eventually equi-right-continuous at .
Thus, it is deduced that the series and define a fuzzy number if the sequences satisfy the conditions of Theorem 8.
Theorem 9 (cf. ). The following statements for level addition of fuzzy numbers and classical addition + of real scalars are valid.(i) is neutral element with respect to ; that is, for all .(ii)With respect to , none of has opposite in .(iii)For any with or , and any , we have . For general , the above property does not hold.(iv)For any and any , we have .(v)For any and any , we have .
2.1. Generalized Hukuhara Difference
Let be the space of nonempty compact and convex sets in the -dimensional Euclidean space . If , denote by the set of (closed bounded) intervals of the real line. Given two elements and , 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 . If , scalar multiplication gives the opposite but, in general, ; 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 for all and for all . 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  aims to overcome this situation.
Definition 10 (see [15, Definition 1]). The generalized Hukuhara difference of two sets is defined as follows:
Proposition 11 (see ). The following statements hold. (a) Let 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.(ii).(iii).(iv) and .(b) The H-difference of two intervals and always exists and where , which hold in Definition 10 are satisfied simultaneously if and only if the two intervals have the same length and .
Proposition 12 (see ). The following statements hold.(a) If and are two closed intervals, then .(b) Let be such that . Then, we havewhere the limits are in the Hausdorff metric for intervals.
3. 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 ). Consider a function from into 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 = = consisting of each of the functions depending on for all . Then, the function is said to be a fuzzy-valued function on for all .
Remark 14. The functions with given in Definition 13 are also defined for all as , where is any constant.
Now, following Kadak , we give the classical sets and consisting of the continuous and bounded fuzzy-valued functions; that is, Obviously, from Representation Theorem, each of the functions which depend on is left continuous on and right continuous at . It was shown that and are complete with the metric on defined by means of the Hausdorff metric as where and are the elements of the sets or with .
3.1. 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 , we can define H-differentiation of a fuzzy-valued function with respect to level sets. For short, throughout the paper, we write instead of “Hukuhara sense.”
Definition 15. A fuzzy-valued function is said to be generalized H-differentiable with respect to the level sets at if (1) exists such that, for all sufficiently near to , the H-difference exists; then the H-derivative is given as follows: or (2) exists such that, for all sufficiently near to , the H-difference exists; then the H-derivative is given as follows: for all and .
From here, we remember that the H-derivative of at depends on the value and the choice of a constant .
Corollary 16. A fuzzy-valued function is H-differentiable if and only if and are differentiable functions in the usual sense.
Definition 17 (periodicity). A fuzzy-valued function is called periodic if there exists a constant for which for any . Thus, it can easily be seen that the conditions and hold for all and . Such a constant is called a period of the function .
3.2. 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 , there exists such that for any division of with the norms , where and , and is -integrable. One can conclude that in (22) denotes the usual Riemann sum for any division of .
Theorem 19 (see [19, Theorem 8.8]). Let and FH-integrable on . If there exists such that , then
Remark 20. We remark that the integrals in (23) exist in the usual sense for all and . It is easy to see that the pair of functions are continuous.
Remark 21. Note that if is periodic fuzzy-valued function and -integrable on any interval of length , then it is -integrable on any other of the same length, and the value of the integral is the same; that is, for all and .
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 -integrable, we mean that it is -integrable on an interval of length . It follows from the property just proved that is also -integrable on any interval of finite length.
Definition 22 (see  (uniform convergence)). Let be a sequence of fuzzy-valued functions defined on a set . We say that converges pointwise on if for each the sequence converges for all and . If a sequence converges pointwise on a set , then we can define by
In other words, converges to on if and only if for each and for an arbitrary , there exists an integer such that whenever . The integer in the definition of pointwise convergence may, in general, depend on both 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 , there exists an integer such that Obviously, the sequence of fuzzy-valued functions converges to a fuzzy valued-function if and only if and converge uniformly to and in , respectively. Often, we say that is the uniform limit of the sequence on and write , , uniformly on .
Now, as a consequence of Definition 22, the following theorem determines the characterization of uniform convergence of fuzzy-valued sequences.
Theorem 23 (see ). Let and . Then, the following statements are valid.(i)A sequence of fuzzy-valued functions defined on a set converges uniformly to a fuzzy-valued function on if and only if (ii)The limit of a uniformly convergent sequence of continuous fuzzy-valued functions on a set is continuous. That is, for each ,
Theorem 24 (interchange of limit and integration). Suppose that for all such that converges uniformly to on . By combining this and the inclusion (28), the equalities hold, where the integral exists for all and . Also, for each , it is trivial that and the convergence is uniform on .
Proof. Note that by Part (ii) of Theorem 23, is continuous on , so that exists. Let be given. Then, since uniformly on , there is an integer such that for . Again, since the distance function is continuous on , it follows 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.
Remark 25. The uniform convergence of is sufficient but is not necessary. In other words the conclusion of Theorem 24 holds without being convergent uniformly on .
Definition 26. The series is said to be uniformly convergent to a fuzzy-valued function on if the partial level sum converges uniformly to on . That is, the series converges uniformly to if, given any , there exists an integer such that for all and whenever .
Corollary 27. If is a continuous fuzzy-valued function on for each and is uniformly convergent to on , then is continuous on for all .
Corollary 28 (interchange of summation and integration). Suppose that is a sequence in and converges uniformly to on . Then, where exists for all and .
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 periodic and functions which is given by for all . We now prove some auxiliary formulas for any positive integer such that . 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 ; that is, the system (36) is orthogonal on every such interval, where .
4. Fourier Series for Fuzzy-Valued Functions of Period
Definition 29. Let be a -periodic fuzzy-valued function on a set . The Fourier series of fuzzy-valued function of period is defined as follows: with respect to the fuzzy coefficients and , which converges uniformly in for all and .
Now, we can calculate the Fourier coefficients , , and with respect to the level sets; that is, . We derive from (38) by FH-integrating over that As an extension of the relation (39) to write with level sets, we have for each and . By taking into account the formulas of orthogonal system in (36) for each with , to get , and by multiplying (39) by , we obtain by FH-integrating it over that
Similarly to get , multiplying (39) by and we present by FH-integrating it over that the coefficients , , and with respect to the level sets are derived that Combining the trigonometric identity 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 .
Therefore, in looking for a trigonometric series of fuzzy-valued 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 -integrating of a fuzzy-valued function of period . Therefore, the interval of integration can be replaced by any other interval of length .
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 .
Example 31. Let be -periodic fuzzy valued function and FH-integrable on with trapezoidal form defined by which is FH-integrable on for each and . By using Definition 1, the level set of the membership function can be written as follows: Therefore, we calculate the fuzzy Fourier coefficients , , and as follows: By considering above coefficients in (38) and the condition if , we have
Definition 32 (complex form). Let be a fuzzy-valued function and -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: and , in (38), the complex form of Fourier series of fuzzy-valued function is given by
where the H-difference exists for all 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:
Definition 33. Let be any fuzzy-valued function on , defined either on the whole -axis or on some intervals. Then, is said to be an even function if for every . Thus, the conditions and hold for all and .
Definition 34. Let be an even function on , or else an even periodic function. Then, the Fourier fuzzy coefficients of are and . Therefore, Fourier series of an consists of ; that is,
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 , then fuzzy Fourier series consists of sines, that is,
Definition 37 (one-sided H-derivatives). Let be any fuzzy-valued 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,
for all . In addition, we suppose that the generalized left-hand H-derivative exists and is defined by
Thus, we can write
If is continuous at , this coincides with the usual left-hand derivative; if has a discontinuity at , we take care to use the left-hand instead of just writing .
Symmetrically, we shall also assume that the generalized right-hand H-derivative 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 ] (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 . That is, and the second property is
Lemma 39. Let and FH-integrable on ; then 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 where and