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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 [

The rest of this paper is organized as follows. In Section

A

The set

We can define trapezoidal fuzzy number

Let

One can see that

Let

The following basic statements hold.

[

[

[

According to Definition

Obviously the sequence

The boundedness of the sequence

If the sequence

Let

Let

Let

Thus, it is deduced that the series

The following statements for level addition

With respect to

For any

For general

For any

For any

Let

To partially overcome this situation, the

A generalization of the Hukuhara difference proposed in [

The generalized Hukuhara difference

The following statements hold.

The following statements hold.

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.

Consider a function

The functions

Now, following Kadak [

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 [

A fuzzy-valued function

From here, we remember that the H-derivative of

A fuzzy-valued function

A fuzzy-valued function

A fuzzy valued function

Let

We remark that the integrals

Note that if

This property is an immediate consequence of the interpretation of an integral as an area. In fact, each integral (

Let

In other words,

Now, as a consequence of Definition

Let

A sequence of fuzzy-valued functions

The limit of a uniformly convergent sequence of continuous fuzzy-valued functions

Suppose that

Note that by Part (ii) of Theorem

The hypothesis of Theorem

The uniform convergence of

The series

If

Suppose that

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

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

Let

Now, we can calculate the Fourier coefficients

Similarly to get

Therefore, in looking for a trigonometric series of fuzzy-valued functions whose level sum is a given fuzzy-valued function

Let

Let

Let

If we set

and then the

Let

Let

By taking into account Definition

Let

Let

Symmetrically, we shall also assume that the generalized right-hand H-derivative

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.

The Dirichlet kernel

Let

By taking into account FH-integration and the Dirichlet kernel defined in Lemma

Suppose that

Let

Consider

Let

the arithmetic mean of the right-hand and left-hand limits

(i) Firstly, continuity and the existence of one-sided H-derivatives are sufficient for convergence. Secondly, if

(ii) The continuity means that Fourier fuzzy coefficients

By taking into account (

We assume that the above results hold with respect to

As conventional hardware systems have been based on membership functions, a membership grade has been assigned to each element in the universe of discourse [

One of the purposes of this work is to extend the classical analysis to the fuzzy level set analysis dealing with fuzzy-valued 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

The authors declare that there is no conflict of interests regarding the publication of this paper.