Fuzzy Hilbert Transform of Fuzzy Functions

Abstract

This paper studies the properties of the Fourier transform of the fuzzy function, and extends the classical Poisson integral formula on the half plane to the fuzzy case, obtaining the composition of the fuzzy set generated by a point in the complex field under the action of the fuzzy function. Further, we define and study the fuzzy Hilbert transform of fuzzy functions and their properties. We prove that when the fuzzy function degenerates to the classical case, the fuzzy Hilbert transform will degenerate to the classical Hilbert transform, which proves that the fuzzy Hilbert transform is an extension of classical transformations in the fuzzy function space. In addition, we point out and prove some properties of the fuzzy Hilbert transform. For some fuzzy functions that meet certain requirements, their fuzzy Hilbert transform is a fuzzy point on 0.

1. Introduction

In 1965, American cybernetics expert L A. Zadeh [1] introduced the concept of fuzzy sets. The birth of the concept of fuzzy sets has expanded many classic mathematical theories to a wider range of fields, such as fuzzy topology [2,3,4,5], fuzzy convex structures [6,7], fuzzy linear spaces [8,9], fuzzy analysis [10], fuzzy measures [11,12], fuzzy integrals [13,14,15], and fuzzy groups [16,17,18,19]. Afterwards, people discovered that fuzzy mathematics can be used to describe the processes of people’s judgment, evaluation, reasoning, decision, and control, which has led to the emergence of a number of highly applicable fuzzy mathematics-related theories, such as fuzzy cluster analysis, fuzzy pattern recognition, fuzzy comprehensive evaluation, fuzzy decision, fuzzy prediction, fuzzy control, and fuzzy information processing.

Recently, research on fuzzy functions in the field of fuzzy mathematics has characterized a method that maps the domain of a classical function to a fuzzy subset of its value range, laying a theoretical foundation for studying some uncertainty problems. In this case, many practical applications can be seen as fuzzy functions. For example, the position of a particle in space at a certain moment is uncertain, and this position can be represented as a fuzzy subset. As another example, when a signal is received with errors, the value of the received signal with errors at a certain moment can be represented as a fuzzy set, and this erroneous signal is a fuzzy function.

Here is a simple and specific case. For a sinusoidal signal, due to certain limitations or interference during transmission, the receiver experiences uncertainty, and the received signal can be regarded as follows: with a degree of 0.6, it is $f\left(t\right)=sin\left(t\right)$, and with a degree of 0.4, it is $f\left(t\right)=sin(t+0.05)$. Then, if we want to retain all information, including uncertainty, we can regard the signal as a fuzzy function and the value of the function at t is $\widetilde{A_t}$, which is a fuzzy subset of the universal set composed of all possible received values expressed as follows:

$$\widetilde{A_t}\left(x\right)=\left\{\begin{array}{ccc}0.6& ,x=& sin\left(t\right)\\ 0.4& ,x=& sin(t+0.05)\end{array},\right.$$

To study such issues, we need to study some further properties of fuzzy functions, such as the Fourier transform, the Hilbert transform, and the composition of the fuzzy subset corresponding to the fuzzy function at a certain moment.

In classical cases, the function transform, such as the Hilbert transform, has a wide range of applications. Here are some examples of the latest practical applications of the classical Hilbert transform: real-time hybrid simulation [20], multiple matching attenuation [21], displacement measurement method [22], individual microscale particle detection [23], nanometer micro-displacement reconstruction [24], etc. Due to the widespread uncertainty and errors in things, the use of fuzzy functions and fuzzy Hilbert transform offers the opportunity to further develop these studies.

Because the range of fuzzy functions is a set of fuzzy subsets, we need to extend these transformations to the fuzzy function space. Compared to some traditional methods, the fuzzy function used in this paper can fully preserve the characterization of uncertainty, providing it with a wider range of practical value.

Regarding the innovation of the article, one point should be noted. The fuzzy functions or fuzzified Fourier transforms mentioned in this paper are not existing concepts, nor are they the fuzzy functions or fuzzy Fourier transforms in some engineering fields. Some concepts just have similar names, but their actual contents are completely different. For specific details, please refer to the detailed definitions given in this paper. Readers are asked to make distinctions.

For example, there are some definitions of fuzzified transforms in the existing literature, such as F-transforms. These definitions have nothing to do with this paper. Specifically, the core idea of the technique of F-transforms is a fuzzy partition of a universe into fuzzy subsets and the consideration of its average values over fuzzy subsets from the partition [25,26]. However, the research in this paper is based on the definition of fuzzy functions that map classical sets to fuzzy sets. The operations of this set-valued function are defined in the paper, and then the subsequent work is carried out by utilizing the properties of this function. Compared with previous formal definitions, the definition method in this paper has more practical application backgrounds and values.

The fuzzy functions defined in this paper remain fuzzy functions after such fuzzified Fourier transforms and fuzzified Hilbert transforms. This property is excellent as it can maintain the closure of the concept of fuzzy functions. In addition, this paper generalizes the Poisson integral formula in the classical case to the fuzzy case, which enables the author to study the composition of the values (where the value is a fuzzy set) of fuzzified functions at a certain point by using relevant properties. This makes the research in this paper more practically valuable.

2. Preliminaries

Definition 1. 

Fuzzy subset. A fuzzy subset $\tilde{A}$ of X defined by

$$\begin{array}{cccc}{\mu }_A:& X& \to & L,\\ & x& \mapsto & {\mu }_A\left(x\right),\end{array}$$

$A\subseteq X$ is a subset of X, and L is a lattice; we say that ${\mu }_A$ determines a fuzzy subset $\tilde{A}$ of X; we can also say that ${\mu }_A$ is a fuzzy subset of X [1].

Definition 2. 

Fuzzy point. Specifically, if

$${\mu }_{A_b}\left(x\right)=\left\{\begin{array}{c}C,x=b\\ 0,x\ne b\end{array},\right.$$

$b\in X$, C is a constant independent of x.

We call the fuzzy subset ${\mu }_{A_b}$ a fuzzy point of X, noted as $P_b$ [27].

Definition 3. 

Fuzzy function. For a set T, construct mapping

$$\begin{array}{ccc}\tilde{f}:& T& \to L^X,\\ & t& \mapsto {\mu }_A,\end{array}$$

we call the mapping $\tilde{f}$ a fuzzy function.

For any classical function f with the form

$$\begin{array}{ccc}f:& T& \to X,\\ & t& \mapsto x_t,\end{array}$$

we can say that $\tilde{f}$ is a fuzzy function of f.

Specifically, if

$${\mu }_A\left(x\right)=\left\{\begin{array}{c}\hfill C,x=x_t\\ \hfill 0,x\ne x_t\end{array},\right.$$

$C\in L$ is a constant independent of $t,x$; then, $\tilde{f}\left(t\right)$ is only related to $x_t$, and it degenerates into the classical function f.

Throughout this paper, we let the classical function $f:T\to X$ be almost everywhere continuous; $\tilde{f}$ is the fuzzy function of f; for $\forall t\in T$, ${\tilde{f}}_t\left(x\right)$ is continuous almost everywhere on X; $L=R$; to avoid ambiguity, we note it as D; and $\int \left|\tilde{f}\right|dx$ is uniformly bounded about t, without loss of generality; let

$$\int \left|\tilde{f}\right|dx=\int \left|f(t,x)\right|dx\le 1,\forall t=t_0,$$

and

$$\underset{t\to +\infty }{lim}f(t,x)=0,$$

$$\underset{t\to -\infty }{lim}f(t,x)=0,$$

uniformly with x, and for $\forall x$,

$${\int }_R{\left|\tilde{f}\right|}^{2}dt<\infty .$$

Definition 4. 

We note that $\mathcal{F}$ consists of all f meeting the above requirements, and $\tilde{\mathcal{F}}$ consists of all $\tilde{f}$ meeting the above requirements; for convenience, we write $\tilde{f}=\tilde{\mathcal{F}}\left(f\right)$.

Definition 5. 

Addition and multiplication of fuzzy functions. For $\forall {\mu }_{A_f}\in D^X$, let

$$\overline{{\mu }_{A_f}}\left(x\right)=\left\{\begin{array}{c}\hfill {\mu }_{A_f}\left(x\right),x\in A\\ \hfill 0,x\notin A\end{array},\right.$$

and let

$$\overline{{\mu }_{A_h}}\left(x\right)=\overline{{\mu }_{A_f}}\left(x\right)+\overline{{\mu }_{A_g}}\left(x\right),$$

$$\overline{{\mu }_{A_k}}\left(x\right)=\overline{{\mu }_{A_f}}\left(x\right)·\overline{{\mu }_{A_g}}\left(x\right).$$

For convenience, we denote $\overline{{\mu }_{A_h}}$ as ${\mu }_{A_h}$ and $\overline{{\mu }_{A_k}}$ as ${\mu }_{A_k}.$ Since $D=R$ is closed for addition and multiplication, we can define $\tilde{f}+\tilde{g}$ and $\tilde{f}·\tilde{g}$ as follows.

$$\begin{array}{ccc}\tilde{f}+\tilde{g}:& T& \to D^X,\\ & t& \mapsto {\mu }_{A_h},\end{array}$$

$$\begin{array}{ccc}\tilde{f}·\tilde{g}:& T& \to D^X,\\ & t& \mapsto {\mu }_{A_k}.\end{array}$$

When we regard $\tilde{f},\tilde{g}$ as functions related to $(t,x)$, we can write

$$\left(\tilde{f}+\tilde{g}\right)(t,x)=\tilde{f}(t,x)+\tilde{g}(t,x),$$

$$\left(\tilde{f}·\tilde{g}\right)(t,x)=\tilde{f}(t,x)·\tilde{g}(t,x).$$

Example 1. 

Let $f\left(t\right)=sin\left(t\right),g\left(t\right)=cos\left(t\right)$ be two classical functions, and $\tilde{f},\tilde{g}$ be their corresponding fuzzy functions.

$$\tilde{f}\left(t_0\right)={\mu }_{A_{f,t_0}},$$

$$\tilde{g}\left(t_0\right)={\mu }_{A_{g,t_0}},$$

$${\mu }_{A_{f,t_0}}\left(x\right)=x^2-2sin\left(t_0\right)x+{sin}^2\left(t_0\right)-1,$$

$${\mu }_{A_{g,t_0}}\left(x\right)=x^2-2cos\left(t_0\right)x+{cos}^2\left(t_0\right)-1.$$

Then,

$${\mu }_{A_{h,t_0}}\left(x\right)=2x^2-2(sin\left(t_0\right)+cos\left(t_0\right))x+{sin}^2\left(t_0\right)+{cos}^2\left(t_0\right),$$

$$\left(\tilde{f}+\tilde{g}\right)\left(t\right)={\mu }_{A_h},$$

$${\mu }_{A_{k,t_0}}\left(x\right)=(x-(sin\left(t_0\right)+1))(x-(sin\left(t_0\right)-1))(x-(cos\left(t_0\right)+1))(x-(cos\left(t_0\right)-1)),$$

$$\left(\tilde{f}·\tilde{g}\right)\left(t\right)={\mu }_{A_k},$$

3. Fuzzy Fourier Transform for Fuzzy Functions

Definition 6. 

Fourier transform of fuzzy function. Let $f\in \mathcal{F}$, and $\tilde{f}\in \tilde{\mathcal{F}}$; in the classical case, the Fourier transform of f is defined as

$$g\left(w\right)=F\left[f\right]\left(t\right)={\int }_{T}f\left(t\right)e^{-jwt}dt.$$

The fuzzy Fourier transform of a fuzzy function is defined by

$$\tilde{F}\left[\tilde{f}\right](w,x)=\tilde{g}(w,x)={\int }_T\tilde{f}(t,x)e^{-jwt}dt.$$

Rationality of Definition 6.

Consider

$$z={\int }_T\tilde{f}{e}^{-jwt}dt,$$

when fixed $w=w_0$, note

$${\mu }_z\left(x\right)={\int }_T\tilde{f}{(t,x)e}^{-j{w}_{0}t}dt,$$

then

$$\begin{array}{cccc}{\mu }_z:& X& \to & D,\\ & x& \mapsto & {\mu }_z\left(x\right),\end{array}$$

${\mu }_z$ is a fuzzy set on X, which means that z is a fuzzy function; furthermore, z is a fuzzy function of $g=F\left[f\right]$. Note that $\tilde{g}=z$. We say that $\tilde{g}$ is the Fourier transform of $\tilde{f}$. Note that $\tilde{F}\left[\tilde{f}\right]=\tilde{g}$.

As above, let V be the set composed of all f, and $s_f$ be a mapping from $\mathcal{F}$ to $\tilde{\mathcal{F}}$, and $s_f\left(f\right)=\tilde{f}$, $\exists s_g$ satisfying

$$\begin{array}{ccc}{s}_g:& \mathcal{F}& \to \tilde{\mathcal{F}},\\ & g& \mapsto \tilde{g}.\end{array}$$

Theorem 1. 

The fuzzy Fourier transform of fuzzy function satisfies the following properties:

(FzFT1) 

Linearity. Let $\widetilde{g_1}=\tilde{F}\left[\widetilde{f_1}\right]$, $\widetilde{g_2}=\tilde{F}\left[\widetilde{f_2}\right]$, $a,b\in D$; then, $\tilde{F}\left[a\widetilde{f_1}+b\widetilde{f_2}\right]=a\widetilde{g_1}+b\widetilde{g_2}$.

(FzFT2) 

Time shifting property. $\tilde{F}\left[\tilde{f}\left(t+t_0,x\right)\right]=\tilde{g}(w,x)e^{jw{t}_0}$.

(FzFT3) 

Frequency shifting property. $\tilde{F}\left[\tilde{f}(t,x)e^{j{w}_{0}t}\right]=\tilde{g}\left(w-w_0,x\right)$.

(FzFT4) 

Scale shifting. $\tilde{F}\left[\tilde{f}(at,x)\right]={\displaystyle \frac{1}{\left|a\right|}}\tilde{g}\left({\displaystyle \frac{w}{a}},x\right)$, $a\ne 0$.

Proof. 

According to Definitions 5 and 6, we have

• Linearity.

  $$\begin{array}{cc}\hfill \tilde{F}\left[a\widetilde{f_1}+b\widetilde{f_2}\right]& ={\int }_T\left(a\widetilde{f_1}+b\widetilde{f_2}\right)(t,x)e^{-jwt}dt\hfill \\ \hfill & ={\int }_T{\left(a\widetilde{f_1}(t,x)+b\widetilde{f_2}(t,x)\right)e}^{-jwt}dt\hfill \\ \hfill & ={\int }_T{a\widetilde{f_1}(t,x)e}^{-jwt}dt+{\int }_T{b\widetilde{f_2}(t,x)e}^{-jwt}dt\hfill \\ \hfill & =a{\int }_T{\widetilde{f_1}(t,x)e}^{-jwt}dt+b{\int }_T{\widetilde{f_2}(t,x)e}^{-jwt}dt\hfill \\ \hfill & =a\widetilde{g_1}+b\widetilde{g_2}.\hfill \end{array}$$
• Time shifting property.

  $$\begin{array}{cc}\hfill \tilde{F}\left[\tilde{f}\left(t+t_0,x\right)\right]& ={\int }_T\tilde{f}\left(t+t_0,x\right)e^{-jwt}dt\hfill \\ \hfill & ={\int }_T\tilde{f}\left(t+t_0,x\right)e^{-jw\left(t+t_0\right)}{e}^{jw{t}_0}d\left(t+t_0\right)\hfill \\ \hfill & =e^{jw{t}_0}{\int }_T{\tilde{f}(u,x)e}^{-jwu}du\hfill \\ \hfill & =\tilde{g}(w,x)e^{jw{t}_0}.\hfill \end{array}$$
• Frequency shifting property.

  $$\begin{array}{cc}\hfill \tilde{F}\left[\tilde{f}(t,x)e^{j{w}_{0}t}\right]& ={\int }_T\tilde{f}(t,x)e^{j{w}_{0}t}{e}^{-jwt}dt\hfill \\ \hfill & ={\int }_T\tilde{f}(t,x)e^{-j\left({w-w}_0\right)t}dt\hfill \\ \hfill & =\tilde{g}\left(w-w_0,x\right).\hfill \end{array}$$
• Scale shifting.

  $$\begin{array}{cc}\hfill \tilde{F}\left[\tilde{f}(at,x)\right]& ={\int }_T\tilde{f}(at,x)e^{-jwt}dt\hfill \\ \hfill & ={\int }_T\tilde{f}(u,x)e^{-jw{\displaystyle \frac{u}{a}}}d{\displaystyle \frac{u}{a}}\hfill \\ \hfill & ={\displaystyle \frac{1}{\left|a\right|}}{\int }_T\tilde{f}(u,x)e^{-j{\displaystyle \frac{w}{a}}u}du\hfill \\ \hfill & ={\displaystyle \frac{1}{\left|a\right|}}\tilde{g}\left({\displaystyle \frac{w}{a}},x\right).\hfill \end{array}$$

□

4. Poisson Integral Formula for Fuzzy Functions

Let $f\left(z\right)$ be a classical analytic function defined on the complex field, satisfying $f\to 0$, $z\to \infty$ on the upper semicircle C of the complex plane. For $\forall x$, we require $\tilde{f}\to 0$, $z\to \infty$ uniformly with x. Let $z_0$ be inside C; $\overline{z_0}$ is the complex conjugate of $z_0$.

Definition 7. 

Consider $\tilde{f}:z\to {\mu }_A$; according to the definition, ${\mu }_A$ can also be regarded as the following set

$${\mu }_A=\left\{\left(x,{\mu }_A\left(x\right)\right)\right\}.$$

Define

$${\mu }_{A_{x_0}}\left(x\right)=\left\{\begin{array}{c}\hfill {\mu }_A\left(x_0\right),x=x_0\\ \hfill 0,x\ne x_0\end{array},\right.$$

and then we have

$${\mu }_A=\bigcup _{x_0\in X}{\mu }_{A_{x_0}}.$$

Define $\widetilde{{}^{\alpha }f}\in \mathcal{F}$ as follows

$$\begin{array}{cccc}\widetilde{{}^{\alpha }f}:& T& \to & D^X,\\ & t& \mapsto & {\mu }_{A_{x_0}},\end{array}$$

and we also require the classical function

$${}^{\alpha }f=\left\{\left({}^{\alpha }t{,}^{\alpha }{x}_0\right)\right\}\in \mathcal{F};$$

due to D being dense,

$$\tilde{f}=\bigcup _{\alpha }\widetilde{{}^{\alpha }f}.$$

Example 2. 

Let f be a classical function and $\tilde{f}$ be the corresponding fuzzy function. Let

$${\mu }_{A_t}\left(x\right)=-{\displaystyle \frac{1}{k}}x+{\displaystyle \frac{f\left(t\right)}{k}}+1,x\in [f\left(t\right),f\left(t\right)+k],k>0.$$

$${\mu }_{A_{t,\alpha }}\left(x\right)=\left\{\begin{array}{c}\hfill -{\displaystyle \frac{1}{k}}\alpha +{\displaystyle \frac{f\left(t\right)}{k}}+1,x=\alpha \\ \hfill 0,x\ne \alpha \end{array},\right.$$

among them, $\alpha \in \left[f\right(t),f(t)+k],k>0.$

$$\widetilde{{}^{\alpha }f}\left(t\right)={\mu }_{A_{t,\alpha }}.$$

Therefore,

$${\mu }_A=\bigcup _{\alpha \in \left[f\right(t),f(t)+k]}{\mu }_{A_{t,\alpha }},k>0.$$

and the relationship between $\widetilde{{}^{\alpha }f}$ and $\tilde{f}$ specified above is as follows

$$\tilde{f}=\bigcup _{\alpha \in \left[f\right(t),f(t)+k]}\widetilde{{}^{\alpha }f},k>0.$$

Theorem 2. 

The composition of $\tilde{f}$ at $z_0$ can be characterized as follows

$$\tilde{f}\left(z_0\right)=\bigcup _{\alpha }\left({\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1\right).$$

Proof. 

Note that ${\forall }^{\alpha }f=\left\{\left({}^{\alpha }t{,}^{\alpha }{x}_0\right)\right\}$, corresponding to only one element in $\tilde{\mathcal{F}}$, i.e., $\widetilde{{}^{\alpha }f}=\left\{\left({}^{\alpha }t,{\mu }_{A_{{}^{\alpha }x_0}}\right)\right\}$, so there is a bijection between $\left\{{}^{\alpha }f\right\}$ and $\left\{\widetilde{{}^{\alpha }f}\right\}$; we note it as $\eta$, i.e.,

$$\eta \left({}^{\alpha }f\right)=\widetilde{{}^{\alpha }f},$$

$${\eta }^{-1}\left(\widetilde{{}^{\alpha }f}\right){=}^{\alpha }f.$$

According to the Cauchy integral formula, for ${\forall }^{\alpha }f\in \mathcal{F}$,

$${}^{\alpha }f\left(z_0\right)={\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{{}^{\alpha }f\left(z\right)}{z-z_0}}dz,$$

and then

$$\widetilde{{}^{\alpha }f}\left(z_0\right)=\eta \left({}^{\alpha }f\left(z_0\right)\right)={\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-z_0}}dz,$$

because of

$$\tilde{f}\left(z_0\right)=\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right);$$

therefore,

$$\tilde{f}\left(z_0\right)=\bigcup _{\alpha }{\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-z_0}}dz.$$

According to Cauchy’s integral theorem,

$${\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{{}^{\alpha }f\left(z\right)}{z-\overline{z_0}}}dz=0.$$

Subtracting from the above equation, we have

$${}^{\alpha }f\left(z_0\right)={\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{{}^{\alpha }f\left(z\right)}{z-z_0}}dz-{\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{{}^{\alpha }f\left(z\right)}{z-\overline{z_0}}}dz;$$

therefore, we have

$$\tilde{f}\left(z_0\right)=\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right)=\bigcup _{\alpha }\left({\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-z_0}}dz-{\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-\overline{z_0}}}dz\right),$$

and then we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-z_0}}dz-{\displaystyle \frac{1}{2\pi j}}{\oint }_C{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{z-\overline{z_0}}}dz\hfill \\ \hfill =& {\displaystyle \frac{1}{2\pi j}}{\oint }_C\widetilde{{}^{\alpha }f}\left(z\right)\left({\displaystyle \frac{1}{z-z_0}}-{\displaystyle \frac{1}{z-\overline{z_0}}}\right)dz\hfill \\ \hfill =& {\displaystyle \frac{z_0-\overline{z_0}}{2\pi j}}{\int }_{-r}^r{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1+{\displaystyle \frac{z_0-\overline{z_0}}{2\pi j}}{\int }_{C_r}{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{\left(z-z_0\right)\left(z-\overline{z_0}\right)}}dz.\hfill \end{array}$$

Therefore,

$$\tilde{f}\left(z_0\right)=\bigcup _{\alpha }\left({\displaystyle \frac{z_0-\overline{z_0}}{2\pi j}}{\int }_{-r}^r{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1+{\displaystyle \frac{z_0-\overline{z_0}}{2\pi j}}{\int }_{C_r}{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z\right)}{\left(z-z_0\right)\left(z-\overline{z_0}\right)}}dz\right).$$

Let $z=r{e}^{j\theta }\in C$; according to the requirements of $\tilde{f}$,

$$\underset{t\to +\infty }{lim}f(t,x)=0,$$

$$\underset{t\to -\infty }{lim}f(t,x)=0,$$

uniformly with x, and then $\exists M_r>0$, s.t.

$$\left|{}^{\alpha }f\left(r{e}^{j\theta }\right)\right|\le M_r;$$

therefore, when $r\to +\infty$,

$$\begin{array}{cc}\hfill \underset{r\to +\infty }{lim}\left|{\int }_{C_r}{\displaystyle \frac{{}^{\alpha }f\left(z\right)}{\left(z-z_0\right)\left(z-\overline{z_0}\right)}}dz\right|& \le \underset{r\to +\infty }{lim}{\int }_0^{\pi }\left|{\displaystyle \frac{{}^{\alpha }f\left(r{e}^{j\theta }\right)·jr{e}^{j\theta }}{\left(r{e}^{j\theta }-z_0\right)\left(r{e}^{j\theta }-\overline{z_0}\right)}}\right|d\theta \hfill \\ \hfill & \le \underset{r\to +\infty }{lim}{\displaystyle \frac{M_r}{r}}{\int }_0^{\pi }\left|{\displaystyle \frac{e^{j\theta }}{\left(e^{j\theta }-\frac{z_0}{r}\right)\left(e^{j\theta }-\frac{\overline{z_0}}{r}\right)}}\right|d\theta \hfill \\ \hfill & \le \underset{r\to +\infty }{lim}{\displaystyle \frac{M_r}{r}}\left|{\int }_0^{\pi }{e}^{-j\theta }\right|d\theta \hfill \\ \hfill & \le \underset{r\to +\infty }{lim}{\displaystyle \frac{M_r}{r}}=0.\hfill \end{array}$$

Therefore,

$$\underset{r\to +\infty }{lim}{}^{\alpha }f\left(z_0\right)={\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{{}^{\alpha }f\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1.$$

Furthermore,

$$\underset{r\to +\infty }{lim}\widetilde{{}^{\alpha }f}\left(z_0\right)={\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1.$$

Then,

$$\underset{r\to +\infty }{lim}\tilde{f}\left(z_0\right)=\underset{r\to +\infty }{lim}\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right).$$

Next, we prove

$$\underset{r\to +\infty }{lim}\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right)=\bigcup _{\alpha }\left({\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1\right).$$

For $\forall \left(x_0,d_0\right)\in \widetilde{A_{z_0}}$, and satisfying

$$\left(x_0,d_0\right)\in \bigcup _{\alpha }\left({\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1\right),$$

$\exists {\alpha }_0$, s.t.

$$\left(x_0,d_0\right)\in {\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{{\alpha }_0}f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1.$$

Then $\exists p_0>0$, s.t. when $p>p_0$,

$$\left(x_0,d_0\right)\in {\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-p}^{+p}{\displaystyle \frac{\widetilde{{}^{{\alpha }_0}f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1.$$

Therefore,

$$\left(x_0,d_0\right)\in \underset{r\to +\infty }{lim}\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right).$$

If

$$\left(x_0,d_0\right)\in \underset{r\to +\infty }{lim}\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right),$$

$\exists r_0>0$, when $r>r_0$,

$$\left(x_0,d_0\right)\in \bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right).$$

Let $\left(x_1,d_1\right)\in \widetilde{{}^{{\alpha }_1}f}$ and $\left(x_2,d_2\right)\in \widetilde{{}^{{\alpha }_2}f}$; according to the definition of $\widetilde{{}^{\alpha }f}$, if $x_1=x_2$, then $d_1=d_2$; if $x_1\ne x_0$ and $x_2\ne x_0$, then $d_1=d_2=0$; else $d_1\ne d_2$.

Then, if $d_0=0$, we have

$$\left(x_0,0\right)\in \underset{r\to +\infty }{lim}\widetilde{{}^{\alpha }f}\left(z_0\right),$$

so we have

$$\left(x_0,0\right)\in \bigcup _{\alpha }\underset{r\to +\infty }{lim}\widetilde{{}^{\alpha }f}\left(z_0\right).$$

If $d_0\ne 0$, then $\exists {\alpha }_0$ s.t. when $r>r_0$,

$$\left(x_0,d_0\right)\in \widetilde{{}^{{\alpha }_0}f}\left(z_0\right),$$

and then

$$\left(x_0,d_0\right)\in \bigcup _{\alpha }\underset{r\to +\infty }{lim}\widetilde{{}^{\alpha }f}\left(z_0\right).$$

In summary,

$$\left(x_0,d_0\right)\in \bigcup _{\alpha }\underset{r\to +\infty }{lim}\widetilde{{}^{\alpha }f}\left(z_0\right).$$

Finally, we have

$$\begin{array}{cc}\hfill \tilde{f}\left(z_0\right)& =\underset{r\to +\infty }{lim}\bigcup _{\alpha }\widetilde{{}^{\alpha }f}\left(z_0\right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{z_{2,0}}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(z_1\right)}{{\left(z_1-z_{1,0}\right)}^2+{z_{2,0}}^2}}d{z}_1\right).\hfill \end{array}$$

□

To summarize, for $z_0$ being inside C, we obtain the composition of the fuzzy set $\widetilde{{}^{\alpha }f}\left(z_0\right)$. In addition, we can consider this as an extension of the Poisson integral formula on a half plane in fuzzy function space.

5. Hilbert Transform for Fuzzy Functions

Definition 8. 

Let $f\in \mathcal{F}$ be a classical function; note that

$$H\left[f\right]\left(t\right)={\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{f\left(\lambda \right)}{t-\lambda }}d\lambda .$$

We call $H\left[f\right]$ the Hilbert transform of f.

Define

$$\tilde{H}\left[\tilde{f}\right]\left(t\right)=\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\lambda }}d\lambda \right);$$

then, for any fixed t, $\tilde{H}\left[\tilde{f}\right]\left(t\right)$ is a fuzzy subset of X. Furthermore, $\tilde{H}\left[\tilde{f}\right]$ is a fuzzy function belonging to $\tilde{\mathcal{F}}$, which means that $\tilde{H}$ is closed on $\tilde{\mathcal{F}}$.

$$\begin{array}{cccc}\tilde{H}:& \tilde{\mathcal{F}}& \to & \tilde{\mathcal{F}},\\ & \tilde{f}& \mapsto & \tilde{H}\left[\tilde{f}\right].\end{array}$$

Specifically, if $\forall t\in T$, $\tilde{f}\left(t\right)=P_{b_t}$, $\tilde{f}$ corresponds to the unique mapping $p_{b_t}$

$$\begin{array}{ccc}{p}_{b_t}:& T& \to X,\\ & t& \mapsto b_t,\end{array}$$

then, $p_{b_t}$ is a classical function. Note that $p_{b_t}$ is just the classical function f, and ${f\left(t\right)=b}_t$. In this case, $\tilde{H}\left[\tilde{f}\right]\left(t\right)$ corresponds to the unique Hilbert transform of classical function $H\left[f\right]\left(t\right)$, so we can regard $\tilde{H}$ as an extension of Hilbert transform on $\tilde{\mathcal{F}}$.

Therefore, we can call $\tilde{H}\left[\tilde{f}\right]$ a fuzzy Hilbert transform of $\tilde{f}$.

Example 3. 

Let $f\left(z\right)=sin\left(z\right)$ be a classical function, $\tilde{f}\left(z\right)={\mu }_A$ be the corresponding fuzzy function, and

$${\mu }_A\left(x\right)=\left\{\begin{array}{c}\hfill -{\displaystyle \frac{1}{sin\left(z\right)}}x+2,sin\left(z\right)\ne 0\\ \hfill 1,sin\left(z\right)=0\end{array},\right.$$

Let

$${\mu }_{A_{\alpha }}\left(x\right)=\left\{\begin{array}{c}\hfill 2-\alpha ,x=\alpha sin\left(z\right)\\ \hfill 0,x\ne \alpha sin\left(z\right)\end{array},\right.\alpha \in [1,2].$$

Note that in the classic case, $H[\alpha sin(z\left)\right]=-\alpha cos\left(w\right)$; then, we have

$$\tilde{H}\left[\widetilde{{}^{\alpha }f}\right]\left(z\right)={\mu }_{B_{\alpha }}.$$

Among them,

$${\mu }_{B_{\alpha }}\left(y\right)=\left\{\begin{array}{c}\hfill 2-\alpha ,y=-\alpha cos\left(\omega \right)\\ \hfill 0,y\ne -\alpha cos\left(\omega \right)\end{array},\right.\alpha \in [1,2].$$

Then, we have,

$${\mu }_B\left(x\right)=\bigcup _{\alpha \in [1,2]}{\mu }_{B_{\alpha }}=\left\{\begin{array}{c}\hfill {\displaystyle \frac{1}{cos\left(\omega \right)}}y+2,cos\left(\omega \right)\ne 0\\ \hfill 1,cos\left(\omega \right)=0\end{array},\right.$$

and we construct the fuzzy function $\tilde{h}$ as follows,

$$\begin{array}{cccc}\tilde{h}:& \Omega & \to & D^Y,\\ & \omega & \mapsto & {\mu }_B.\end{array}$$

Then, $\tilde{h}$ is the Hilbert transform of the fuzzy function $\tilde{f}$, and we note that

$$\tilde{H}\left[\tilde{f}\right]\left(t\right)=\tilde{h}\left(\omega \right)={\mu }_B.$$

6. The Properties of Hilbert Transform of Fuzzy Functions

Theorem 3. 

For $\tilde{f},\widetilde{{}^{\alpha }f}\in \tilde{\mathcal{F}}$, $I_1$ is an indicator set and satisfies

$$\tilde{f}=\bigcup _{\alpha \in I_1}\widetilde{{}^{\alpha }f},$$

if $\exists I_2\subseteq I_1$, it satisfies

$$\left\{\widetilde{{}^{\alpha }f}|\alpha \in I_2\right\}\subseteq \left\{\widetilde{{}^{\alpha }f}|\alpha \in I_1\right\};$$

s.t.

$$\tilde{f}=\bigcup _{\alpha \in I_2}\widetilde{{}^{\alpha }f};$$

and for $\forall \widetilde{{}^{\alpha }f}\in \left\{\widetilde{{}^{\alpha }f}|\alpha \in I_2\right\}$, the corresponding classical function

$${\eta }^{-1}\left(\widetilde{{}^{\alpha }f}\left(t\right)\right){=}^{\alpha }f\left(t\right)=c_{\alpha }.$$

Among them, $c_{\alpha }$ is a constant function independent of t, and then $\tilde{H}\left[\tilde{f}\right]\left(t\right)$ is a fuzzy point at 0.

Proof. 

For $\forall \alpha \in I_2$, according to the conditions, $\exists P_{\alpha }$ s.t.

$$\widetilde{{}^{\alpha }f}\left(t\right)=P_{\alpha };$$

due to $\widetilde{{}^{\alpha }f}$ corresponding to the unique $c_{\alpha }$, in the sense of the Cauchy principal value integral,

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\lambda }}d\lambda & \stackrel{{\eta }^{-1}}{\leftrightarrow }{\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{c_{\alpha }}{t-\lambda }}d\lambda \hfill \\ \hfill & ={\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{c_{\alpha }}{\lambda }}d\lambda \hfill \\ \hfill & ={\displaystyle \frac{c_{\alpha }}{\pi }}\underset{\epsilon \to 0}{lim}\left({\int }_{-\infty }^{-\epsilon }{\displaystyle \frac{1}{\lambda }}d\lambda +{\int }_{+\epsilon }^{+\infty }{\displaystyle \frac{1}{\lambda }}d\lambda \right)\hfill \\ \hfill & ={\displaystyle \frac{c_{\alpha }}{\pi }}\underset{\epsilon \to 0}{lim}\left(-{\int }_{+\epsilon }^{+\infty }{\displaystyle \frac{1}{\lambda }}d\lambda +{\int }_{+\epsilon }^{+\infty }{\displaystyle \frac{1}{\lambda }}d\lambda \right)\hfill \\ \hfill & =0\hfill \\ \hfill & \stackrel{\eta }{\leftrightarrow }{P}_{0,\alpha }.\hfill \end{array}$$

Then, we have

$$\tilde{H}\left[\tilde{f}\right]\left(t\right)=\bigcup _{\alpha \in I_2}\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\lambda }}d\lambda \right)=\bigcup _{\alpha \in I_2}{P}_{0,\alpha }.$$

Let $P_{0,\alpha }$ be

$$P_{0,\alpha }=\left\{\begin{array}{c}{b}_{\alpha },x=0\\ 0,x\ne 0\end{array},\right.$$

and because $P_{0,\alpha }\in D^X$ and D is a lattice, $\exists b^u\in D$, s.t

$$\bigcup _{\alpha \in I_2}{b}_{\alpha }=b^u.$$

Then, ${\bigcup }_{\alpha \in I_2}{P}_{0,\alpha }$ is a fuzzy point at 0 noted as $P_{0,u}$,

$$P_{0,u}=\left\{\begin{array}{c}{b}^u,x=0\\ 0,x\ne 0\end{array}.\right.$$

In summary, $\tilde{H}\left[\tilde{f}\right]\left(t\right)=P_{0,u}$ is a fuzzy point at 0. □

Theorem 4. 

For $\forall k_1,k_2\in D$, $\tilde{f},\tilde{g}\in \tilde{\mathcal{F}}$,

$$\tilde{H}\left[k_1\tilde{f}+k_2\tilde{g}\right]\left(t\right)=k_1\tilde{H}\left[\tilde{f}\right]\left(t\right)+k_2\tilde{H}\left[\tilde{g}\right]\left(t\right).$$

Proof. 

Suppose $\tilde{f}\left(t\right)={\mu }_{A_f}$, $\tilde{g}\left(t\right)={\mu }_{A_g}$; then,

$$\left(k_1\tilde{f}+k_2\tilde{g}\right)\left(t\right)=k_1\tilde{f}\left(t\right)+k_2\tilde{g}\left(t\right)$$

$$=k_1{\mu }_{A_f}+k_2{\mu }_{A_g};$$

for $\forall x$,

$$\left(k_1{\mu }_{A_f}+k_2{\mu }_{A_g}\right)\left(x\right)=k_1{\mu }_{A_f}\left(x\right)+k_2{\mu }_{A_g}\left(x\right);$$

due to D being a ring,

$$k_1{\mu }_{A_f}\left(x\right)+k_2{\mu }_{A_g}\left(x\right)\in D,$$

and then $\exists \tilde{h}=k_1\tilde{f}+k_2\tilde{g}\in \tilde{\mathcal{F}}$.

Let

$$\tilde{h}=\bigcup _{\alpha }\widetilde{{}^{\alpha }h};$$

due to

$${\mu }_{A_h}\left(x\right){=k}_1{\mu }_{A_f}\left(x\right)+k_2{\mu }_{A_g}\left(x\right),$$

we have

$$\widetilde{{}^{\alpha }h}\left(\lambda \right)=k_1\widetilde{{}^{\alpha }f}\left(\lambda \right)+k_2\widetilde{{}^{\alpha }g}\left(\lambda \right),$$

and thus

$$\begin{array}{cc}\hfill \tilde{H}\left[k_1\tilde{f}+k_2\tilde{g}\right]\left(t\right)& =\tilde{H}\left[\tilde{h}\right]\left(t\right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }h}\left(\lambda \right)}{t-\lambda }}d\lambda \right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{k_1\widetilde{{}^{\alpha }f}\left(\lambda \right)+k_2\widetilde{{}^{\alpha }g}\left(\lambda \right)}{t-\lambda }}d\lambda \right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{k_1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\lambda }}d\lambda +{\displaystyle \frac{k_2}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }g}\left(\lambda \right)}{t-\lambda }}d\lambda \right)\hfill \\ \hfill & =k_1\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\lambda }}d\lambda \right)+k_2\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }g}\left(\lambda \right)}{t-\lambda }}d\lambda \right)\hfill \\ \hfill & =k_1\tilde{H}\left[\tilde{f}\right]\left(t\right)+k_2\tilde{H}\left[\tilde{g}\right]\left(t\right).\hfill \end{array}$$

□

Theorem 5. 

For $\forall t_0\in t$, $\tilde{H}\left[\tilde{f}\left(t-t_0\right)\right]=\tilde{H}\left[\tilde{f}\right]\left(t-t_0\right)$.

Proof. 

$$\begin{array}{cc}\hfill \tilde{H}\left[\tilde{f}\left(t-t_0\right)\right]& =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-t_0-\lambda }}d\lambda \right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}\left(\lambda \right)}{t-\left(t_0+\lambda \right)}}d\left(t_0+\lambda \right)\right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}(\lambda +t_0-t_0)}{t-\left(t_0+\lambda \right)}}d\left(t_0+\lambda \right)\right)\hfill \\ \hfill & =\bigcup _{\alpha }\left({\displaystyle \frac{1}{\pi }}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\widetilde{{}^{\alpha }f}(\lambda -t_0)}{t-\lambda }}d\lambda \right)\hfill \\ \hfill & =\tilde{H}\left[\tilde{f}\right]\left(t-t_0\right).\hfill \end{array}$$

□

7. Conclusions

The third section of this article demonstrates that the fuzzy function defined according to Definitions 3–5 in this paper can undergo the fuzzy Fourier transform in Definition 6. The fuzzy function described in Definitions 3–5 remains a fuzzy function after the fuzzy Fourier transform. Moreover, in the rationality of Definition 6, the rationality of this generalization is explained. Theorem 1 indicates that this generalization has properties formally similar to those of the classical Fourier transform, even though what they generate is not a crisp element or a crisp function. Theorem 2 in the fourth section provides a method to characterize the composition of the fuzzy set corresponding to the fuzzy function at a certain point. In Section 5, this article presents the fuzzy Hilbert transform of the fuzzy function corresponding to Definitions 3–5. The fuzzy function described in Definitions 3–5 also remains a fuzzy function after the fuzzy Hilbert transform. Example 3 is given to illustrate that this theory is feasible in practical operation and application. In Section 6, some of their properties formally similar to those in the classical case are presented. In conclusion, the ideas of the fuzzy Fourier transform and the fuzzy Hilbert transform proposed in this paper for fuzzy functions are feasible, and the examples given in this paper demonstrate that this method can indeed be put into practical operation.