A note on fuzzy best approximation using Chebyshev’s polynomials

1 Introduction

The problem of best polynomial approximation is a historical problem in applied mathematics. Since the best polynomial approximation problems appear in many different branches, it is important to study this problem from various view points. In this paper, the fuzzy best Chebyshev approximation is discussed. Combining the results in Powell (1981, Chapter 7) and the work of Abbasbany et al. (2007), we intend to introduce the best Chebyshev approximation (best near-minimax approximation) for fuzzy-valued functions. To do this, we use Chebyshev’s nodes and naturally fuzzy Chebyshev’s polynomials to introduce and compute fuzzy best near-minimax approximation.

The structure of the present paper is as follows. In Section 2, the basic concepts of our work are introduced. Section 3 contains a brief description of Chebychev’s nodes, polynomials and approximation. In Section 4, we introduce the fuzzy best near-minimax approximation and a method for obtaining it. In Section 5, For a case study, the fuzzified Runge phenomenon is studied in details. Finally, we have come to conclusion in Section 6.

2 Preliminaries

Here, we recall the basic notations and concepts for fuzzy triangular numbers, fuzzy-valued functions and fuzzy-valued polynomials.

Let $F(\mathbb{R})$ be the set of all fuzzy numbers, and $\mathit{TF}(\mathbb{R})$ be the set of all triangular fuzzy numbers with membership function $${\mu }_{\tilde{a}}(x)=\left\{\begin{array}{cc}1-\frac{a_s-x}{a_l}\text{,}\hfill & (a_s-a_l)⩽x⩽a_s\text{,}\hfill \\ 1+\frac{a_s-x}{a_r}\text{,}\hfill & a_s⩽x⩽(a_s+a_r)\text{,}\hfill \\ 0\text{,}\hfill & \mathit{otherwise}\text{,}\hfill \end{array}\right.$$ and denote by $\tilde{a}=(a_s\text{,}{a}_l\text{,}{a}_r)$ , where $a_l$ , $a_r$ , and $a_s$ are non-negative real numbers as the left, right spread, and center of the fuzzy number, respectively (Dubois and Prade, 1980).

Also, $\alpha$ -cut of a fuzzy number $\tilde{a}$ is defined by Dubois and Prade (1980) and Abbasbany et al. (2007) $${[\tilde{a}]}^{\alpha }=\left\{\begin{array}{cc}\left\{t\in \mathbb{R}|{\mu }_{\tilde{a}}(t)⩾\alpha \right\}\hfill & \alpha >0\text{,}\hfill \\ \hfill \\ \stackrel{‾}{\left\{t\in \mathbb{R}|{\mu }_{\tilde{a}}(t)>\alpha \right\}}\hfill & \alpha =0\text{.}\hfill \end{array}\right.$$

Let $\tilde{a}=(a_s\text{,}{a}_l\text{,}{a}_r)\text{,}\tilde{b}=(b_s\text{,}{b}_l\text{,}{b}_r)\in \mathit{TF}(\mathbb{R})$ . Some results of applying fuzzy arithmetic on fuzzy numbers $\tilde{a}$ and $\tilde{b}$ are as follows:

• –

  $k>0\text{,}k.\tilde{a}=({\mathit{ka}}_s\text{,}{\mathit{ka}}_l\text{,}{\mathit{ka}}_r)$ ,

• –

  $k<0\text{,}k.\tilde{a}=({\mathit{ka}}_s\text{,}-{\mathit{ka}}_r\text{,}-{\mathit{ka}}_l)$ ,

• –

  $\tilde{a}+\tilde{b}=(a_s+b_s\text{,}{a}_l+b_l\text{,}{a}_r+b_r)$ ,

• –

  $\tilde{a}-\tilde{b}=(a_s-b_s\text{,}{a}_l+b_r\text{,}{a}_r+b_l)$ .

A fuzzy number $\tilde{a}=(a_s\text{,}{a}_l\text{,}{a}_r)\in \mathit{TF}(\mathbb{R})$ is nonnegative if and only if $a_s-a_l⩾0$ , and this fuzzy number is non-positive, if and only if $a_s+a_r⩽0$ .

Denote by ${\prod }_N$ , the set of all real-valued polynomials of degree at most N, and by ${\prod }_N^{+}$ , the set of all nonnegative real-valued piecewise polynomials of degree at most N.

In this study, we employ a class of fuzzy valued polynomials on $\mathit{TF}(\mathbb{R})$ , and we approximate a fuzzy function $\tilde{f}:\mathbb{R}\to \mathit{TF}(\mathbb{R})$ , by such fuzzy valued polynomials (Abbasbany et al., 2007; Kaleva (1994)).

For each $\alpha \in [0\text{,}1]$ , the lower and upper spreads of a fuzzy function $\tilde{f}$ , on its $\alpha$ -cut, are ${[\tilde{f}]}_{-}^{\alpha }$ and ${[\tilde{f}]}_{+}^{\alpha }$ , respectively, such that, for all $x\in \mathbb{R}$ $$\begin{array}{cc}\hfill & {\left[\tilde{f}\right]}_{-}^{\alpha }(x)=\inf \left\{t\in \mathbb{R}|t\in {\left[\tilde{f}(x)\right]}^{\alpha }\right\}\text{,}\hfill \\ \hfill & {\left[\tilde{f}\right]}_{+}^{\alpha }(x)=\sup \left\{t\in \mathbb{R}|t\in {\left[\tilde{f}(x)\right]}^{\alpha }\right\}\text{.}\hfill \end{array}$$

The set of all fuzzy valued polynomials is denoted by ${\prod ^{\sim }}_N$ (Abbasbany et al., 2007; Kaleva (1994)).

3 Chebyshev approximation

We now turn our attention to polynomial interpolation for $f(x)$ over $[-1\text{,}1]$ based on the nodes $-1⩽x_0⩽x_1⩽\cdots ⩽x_N⩽1$ . Let $p_N(x)$ be the Lagrange interpolation polynomial of $f(x)$ (Mathew and Fink, 2004). Therefore, we have $$f(x)=p_N(x)+E_N(x)\text{.}$$ Here, the error function is $E_N(x)=Q(x)\frac{f^{(N+1)}(x)}{(N+1)!}$ , where $Q(x)=(x-x_0)(x-x_1)\cdots (x-x_N)$ . It is well-known that $$\left|E_N(x)\right|⩽\left|Q(x)\right|\frac{\underset{-1⩽x⩽1}{\max }\left\{\left|f^{(N+1)}(x)\right|\right\}}{(N+1)!}\text{.}$$ Chebyshev studied how to minimize the upper bound for $|E_N(x)|$ . One upper bound can be formed by taking the product of the maximum value of $|Q(x)|$ on $[-1\text{,}1]$ and the maximum value $\left|\frac{f^{(N+1)}(x)}{(N+1)!}\right|$ on $[-1\text{,}1]$ . To minimize the statement ${\max }_{-1⩽x⩽1}|Q(x)|$ , Chebyshev discovered that $x_0\text{,}{x}_1\text{,}\dots \text{,}{x}_N$ should be chosen so that $Q(x)=(\frac{1}{2^N})T_{N+1}(x)$ , where $T_{N+1}(x)$ is the polynomial which is generated by $N+1$ Chebyshev’s nodes. Moreover, we have the following well-known theorems and corollary (Philips and Taylor (1980); Powell (1981)).

4 The best near-minimax approximation of fuzzy-valued functions

In this section, we follow Abbasbany et al. (2007) and introduce the best minimax approximation of a fuzzy-valued function on a finite set of distinct points.

Let $\chi =\{x_0\text{,}{x}_1\text{,}{x}_2\text{,}\dots \text{,}{x}_N\}$ be a set of $N+1$ distinct points of $\mathbb{R}$ , and ${\tilde{f}}_i=(f_{i_s}\text{,}{f}_{i_l}\text{,}{f}_{i_r})\in \mathit{TF}(\mathbb{R})$ be the value of a fuzzy function $\tilde{f}:\mathbb{R}\to \mathit{TF}(\mathbb{R})$ at the point $x_i$ , for $i=0\text{,}1\text{,}2\text{,}\dots \text{,}N$ .

Suppose that $\tilde{\beta }=({\beta }_s\text{,}{\beta }_l\text{,}{\beta }_r)$ is a fuzzy number, where ${\beta }_s⩾0$ , and $\Vert \tilde{\beta }\Vert ={\max }_{i=0\text{,}1\text{,}2\text{,}\dots \text{,}N}D(\tilde{p}(x_i)\text{,}{\tilde{f}}_i)$ , that is $${\beta }_s+{\beta }_l+{\beta }_r=\max \left\{|p\left(x_i\right)-f_{i_s}|+|\underline{p}\left(x_i\right)-f_{i_l}|+|\bar{p}\left(x_i\right)-f_{i_r}|\right\}\text{.}$$ Taking

Therefore, we get three independent linear programming problems to be solved:

Solving 3, 2 and 4, we have three independent polynomials $p\text{,}\underline{p}$ and $\bar{p}$ of degree at most N. But, applying 5, right and left spreads of fuzzy valued polynomial may be piecewise polynomials of degree at most N.

Now, we are ready to state the following theorems which may play dominant roles in the best approximation on Chebyshev’s nodes.

5 Runge’s phenomenon

In the field of numerical analysis, Runge’s phenomenon occurs when the polynomials with high degree are applied to approximate the Runge function (Runge, 1901). It was discovered by Runge (1901) when he studied the behavior of errors in the approximation of a certain functions by polynomials. Consider the function (Runng’s function)

The oscillation can be minimized by using Chebyshev nodes instead of the equidistant nodes. In this case, the maximum error is guaranteed to diminish with increasing the degree of polynomial.

In the following, we study this phenomenon numerically when the function $f(x)$ is a fuzzy-valued function. Consider the fuzzy-valued function

Now, the best approximation of function $\tilde{f}(x)$ is computed in two different cases for $N=6$ .

• Case 1.

  Equidistant nodes

  Let $x_i=-1+\frac{i}{3}\text{,}i=0\text{,}1\text{,}\dots \text{,}6$ . Using (2)–(4), the best approximation, ${\stackrel{\sim }{P}}_6=\left(p_6\text{,}{\bar{p}}_6\text{,}{\underline{p}}_6\right)$ , is obtained as follows

  (9)

  $$\begin{array}{c}{f}_s(x)\approx p_6(x)=-8.99x^6+14.7415x^4-6.6698x^2+1\text{,}\hfill \\ f_l(x)\approx {\underline{p}}_6(x)=-4.4974x^6+7.3707x^4-3.3349x^2+0.5\text{,}\hfill \\ f_r(x)\approx {\bar{p}}_6(x)=-4.4974x^6+7.3707x^4-3.3349x^2+0.5\text{.}\hfill \end{array}$$

  The maximum absolute error, i.e. ${\max }_{-1⩽x⩽1}\Vert \tilde{f}(x)-{\stackrel{\sim }{P}}_6(x)\Vert$ , is 1.1877. The results are illustrated in Fig. 1.

• Case 2.

  Chebyshev nodes

  Let $x_i=\cos \left(\frac{(2i+1)\pi }{12}\right)\text{,}i=0\text{,}1\text{,}\dots \text{,}6$ . Using (2)–(4), the best approximation, ${\stackrel{\sim }{Q}}_6=(q_6\text{,}{\bar{q}}_6\text{,}{\underline{q}}_6)$ , is obtained as follows

  (10)

  $$\begin{array}{cc}\hfill & f_s(x)\approx q_6(x)=-5.127x^6+9.4006x^4-5.27x^2+1\text{,}\hfill \\ \hfill & f_l(x)\approx {\underline{q}}_6(x)=-2.5638x^6+4.7003x^4-2.635x^2+0.5\text{,}\hfill \\ \hfill & f_r(x)\approx {\bar{q}}_6(x)=-2.5638x^6+4.7003x^4-2.635x^2+0.5\text{.}\hfill \end{array}$$

  In this case, the maximum absolute error, i.e. ${\max }_{-1⩽x⩽1}\Vert \tilde{f}(x)-{\stackrel{\sim }{Q}}_6(x)\Vert$ , is 0.4040. The results are illustrated in Fig. 2.

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Figure 1

Results for Case 1.

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Figure 2

Results for Case 2.

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6 Conclusions

In this study, we proposed a method for finding the best near-minimax approximation of a fuzzy-valued function on a set of Chebyshev nodes. The existence and uniqueness of the best near-minimax approximation of a fuzzy-valued function were also proved. The best near-minimax approximation was compared to the best approximation (Abbasbany et al., 2007) by Runge’s phenomenon in the fuzzy sense.