Approximation factor of the piecewise linear functions in Mamdani fuzzy system and its realization process 1

Abstract

Piecewise linear function (PLF) is not only a generalization of univariate segmented linear function in multivariate case, but also an important bridge to study the approximation of continuous function by Mamdani and Takagi-Sugeno fuzzy systems. In this paper, the definitions of the PLF and subdivision are introduced in the hyperplane, the analytic expression of PLF is given by using matrix determinant, and the concept of approximation factor is first proposed by using m-mesh subdivision. Secondly, the vertex coordinates and their changing rules of the n-dimensional small polyhedron are found by dividing a three-dimensional cube, and the algebraic cofactor and matrix norm of corresponding determinants of piecewise linear functions are given. Finally, according to the method of solving algebraic cofactors and matrix norms, it is proved that the approximation factor has nothing to do with the number of subdivisions, but the approximation accuracy has something to do with the number of subdivisions. Furthermore, the process of a specific binary piecewise linear function approaching a continuous function according to infinite norm in two dimensions space is realized by a practical example, and the validity of PLFs to approximate a continuous function is verified by t-hypothesis test in Statistics.

Keywords

Piecewise linear function mesh subdivision approximation factor Mamdani fuzzy system matrix norm

1 Introduction

The core of fuzzy system is to bypass the precise mathematical model to carry out logical reasoning and calculation for fuzzy information. The main method is to process data information and language information based on a set of If -Then rules. Generally, Mamdani and T-S fuzzy systems are two kinds of common models, in which Mamdani fuzzy system is the simplest kind of model, the main characteristic of which is that the output of each rule is a fuzzy set, while the output of T-S fuzzy system is a multivariate linear function of the input variables. Although it does not depend on the accurate mathematical model, it has better logical reasoning and numerical calculation and nonlinear function approximation ability. In the late 1990 s, fuzzy systems as approximators have been widely used in the fields of system identification, pattern recognition, nonlinear system design and fuzzy control. See Ref. [1 –3]. Especially in 1998, Ying [4] used the linear programming method to study the general approximation of T-S fuzzy system, and then Zeng [5] gave the sufficient conditions for the approximation of the fuzzy system. These results provide some new ideas and methods for further study of the approximation performance of generalized fuzzy systems.

In 2018, Li and Qin discussed the robustness and general approximation ability of fuzzy reasoning system based on the method of quintuple implication principle (QIP) in [6]. The results show that the QIP method has good performance in the fuzzy system composed of fuzzy rules. In 2019, Kwak et al. [7] proposed a compensatory fuzzy reasoning method based on the movement and deformation between the antecedent fuzzy set and the observation information, which effectively improved the accuracy and learning performance of the algorithm. In 2020, Aubry et al. [8] Put forward an effective algorithm for real solutions of polynomial systems with symmetric L-R fuzzy numbers based on fuzzy systems, and applied it to the computer algebra software package SageMath. Zhang and Hao et al. [9] proposed a multi task genetic fuzzy system based on evolutionary thought and fuzzy optimization, and designed a multi task evolutionary optimization algorithm for Mamdani fuzzy system based on completely overlapping triangular membership function. In 2021, Li and Galayko [10] studied the inverse problem of the isometric feature mapping (ISOMAP) by using multi-level adaptive neuro fuzzy inference system, which effectively solved the potential nonlinear relationship between the low performance results of general ISOMAP reconstruction algorithm and its original data. Setayandeh and Babaei [11] suggested an optimization algorithm based on genetic algorithm for multi-objective constrained optimization problems, and simulated the algorithm by Monte Carlo simulation method. These recent works greatly broaden the application scope of fuzzy system.

The essence of a piecewise linear function is an extension of a segmented linear function of one variable in the case of multivariate variables. It can not only approximate an unknown continuous function with arbitrary precision, but also play an important role in the approximation theory of fuzzy system. In 2000, the concept of multivariate piecewise linear function based on the subdivision input space was first proposed by Prof. Liu in [12], and it is an important bridge to study the approximation of Takagi-Sugeno fuzzy system to continuous function and integrable function in [13]. Subsequently, in 2001, he studied the approximation performance of generalized Mamdani fuzzy system to a class of p-integrable functions in [14], for further information. See Ref. [15 –17]. In 2006, Zhang and Li [18] proved that Mamdani fuzzy system is a universal approximator of integrable function by means of a square piecewise linear function, and provided the necessary conditions for Mamdani fuzzy system to be an approximator in [19]. In recent years, the convergence of fuzzy transformation and the solution of fuzzy dual complex linear systems have been studied. See [20, 21]. However, these results only take PLFs as a bridge to complete the proof, and there is no specific method to obtain it, which naturally limits the wide application of piecewise linear functions.

In 2014, Wang proved the universal approximation of Mamdani fuzzy system by introducing the piecewise linear function in [22]. In the same year, Peng [23] further gave the construction method and analytical expression of the PLF on the basis of Ref. [22], and proposed the solution formula of corresponding equation system through the matrix determinant. In 2015, Tao and Wang et al. [24] introduced K-quasi-subtraction operation to give the concept of Kp-integral norm, and then discussed the approximation of PLFs to a class of integrable functions in 2015, she utilized the piecewise linear functions as a tool to explore the approximation performance of generalized Mamdani fuzzy system to Kp-integrable function in [25]. See [26 , 30]. In 2017, Wang et al. [27] first proposed that the PLFs can approximate a continuous function to arbitrary approximation accuracy in the sense of maximum norm based on the mesh subdivision of a generalized cube. These beneficial works extend the application range of Mamdani fuzzy system from different aspects. In 2021, Wang [28] put forward a new Kp-norm by introducing the K-quasi-decreasing operator and the determinant representation of PLFs, it was proved that Mamdani fuzzy system can approach the PLFs and p-integrable function, and an upper bound representation of subdivision was given. Unfortunately, the authors only guessed that the approximation factor is a constant independent of the number of subdivision in [27], its proof process is not given. This conclusion is the key to whether the final Mamdani fuzzy system can approach a multivariate continuous function.

The main purpose of this article is to deal with and solve the remaining problems in Ref. [27], and to verify that PLFs can approximate continuous functions with any precision by t- hypothesis test in Statistics. The specific solution is to prove that the approximation factor has nothing to do with the number of subdivisions m by using the properties of the coefficient determinants of PLFs, so as to lay a theoretical foundation for Mamdani fuzzy system to approach a continuous function. There are two innovations in this paper. One is to utilize the m- mesh subdivision method to propose the concept of the approximation factor, and then give the representation of vertex coordinates of each generalized small cube by subdividing three-dimensional cube; The other is to prove that the approximation factor is independent of the number of subdivision m by directly calculating the algebraic cofactors and matrix norms, it is only related to the dimension n of input space, the advantage of this method is to get the analytic expression of PLFs.

The remainder of the paper is organized as follows. In Section 2, according to Ref. [14], the concepts of the piecewise linear function and subdivision number are introduced. Meanwhile, we provide the analytical expressions of coefficient matrixes of corresponding linear equation system. In Section 3, the concept of the approximation factor based on the piecewise linear functions is first proposed by applying m-mesh subdivision of a generalized cube, and the calculation method of algebraic cofactors and matrix norm for the corresponding determinant are given. In Section 4, we demonstrate that the approximation factor of a piecewise linear function is independent of the selection of subdivision number by solving algebraic cofactors and matrix norm. In Section 5, the realization process of the binary piecewise linear function approaching a continuous function is given by an example analysis, and it is verified that the approximation factor is independent of the number of subdivision, but the approximation accuracy is related to the number of subdivision.

2 Piecewise linear functions

A piecewise linear function is a generalization of a segmented linear function of one variable in the case of multivariate variables, so it has many excellent properties such as zero outside the compact set of $ℝ^{n}$ , uniformly continuous on the compact set, existence of unilateral partial derivative and bounded. Next, we will give some related concepts of n-variables piecewise linear functions, where the word “n-variables” can be omitted.

In this paper, we use the symbols $ℝ^{n}$ and $ℕ$ to represent the n-dimensional Euclidean space and the set of natural numbers, respectively. For any a > 0, let $\begin{matrix} Δ (a) = {(x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \in ℝ^{n} | 0 \leq x_{i} \leq a, \\ i = 1, 2, \cdot \cdot \cdot, n}, \end{matrix}$ then Δ (a) is called the generalized cube with side length a in $ℝ^{n}$ , and it is abbreviated to Δ (a) = [0, a] × [0, a] × ·· · × [0, a].

Definition 2.1. ^{[22, 27]} Let an n- variables continuous function $S : ℝ^{n} \to ℝ$ . If the following conditions ① to ② are satisfied:

There is a real number a > 0 such that S is always zero outside the generalized cube Δ (a);

If there is a group of n-dimensional polyhedrons {Δ₁, Δ₂, ·· · , Δ_{N
_s}} ⊂ Δ (a) with $\cup_{j = 1}^{N_{s}} Δ_{j} = Δ (a)$ , such that S takes n- variables linear function on each small polyhedron Δ_j (j = 1, 2, ·· · , N_s), that is to say, the S can be expressed as

\begin{matrix} S (x) = \sum_{i = 1}^{n} β_{i j} \cdot x_{i} + λ_{j}, for all \\ x = (x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \in Δ_{j}, j = 1, 2, \cdot \cdot \cdot, N_{s} . \end{matrix}

Then S is called a piecewise linear function on $ℝ^{n}$ , where β_1i1j and λ_j are constants, i = 1, 2, ·· · , n.

In fact, the piecewise linear functions play an important role in studying the approximation of Mamdani fuzzy system and T-S fuzzy system. This is because the fuzzy system can approximate some piecewise linear function, and the piecewise linear function can approximate a continuous or integrable function, thus achieving the fuzzy system approximating to some unknown continuous function.

Definition 2.2. Let $b = (b_{1}, b_{2}, \cdot \cdot \cdot, b_{n}) \in ℝ^{n}$ , $Ω (β) = {x = (x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \in ℝ^{n} | < x, b > = β}$ , then Ω (β) is call a β- hyperplane on $ℝ^{n}$ , where real number $β \in ℝ$ , symbol <· > is inner product. Clearly, the hyperplane Ω (β) can also be expressed in the form of linear combination of multivariate variables, that is, $Ω (β) = {(x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \in ℝ^{n} | b_{1} x_{1} + b_{2} x_{2} + \cdot \cdot \cdot + b_{n} x_{n} = β, b_{i} \in ℝ, i = 1, 2, \cdot \cdot \cdot, n}$ .

Definition 2.3. Let input space be Δ (a) = [0, a] ⁿ (a > 0), and Δ (a) is divided into m_i small closed intervls along each axis x_i (i = 1, 2, ·· · , n) in turn, [0, a] is divided into [0, a/ m_i] , [a/ m_i, 2a/ m_i], ⋯, [a (m_i - 2)/ m_i, a (m_i - 1)/ m_i] , [a (m_i - 1)/ m_i, a]. If all subdivision points can be uniformly listed as $t_{j}^{i} = ja / m_{i}$ , j = 1, 2, ·· · , m_i, then m_i is called an isometry number of subdivision of the axis x_i in the input space Δ (a), also referred to as m_i is the subdivision number on the x_i axis.

Note 1. For simplicity, this paper always assumes that the same number of subdivisions is taken on each axis x_i, i.e., m_i = m, i = 1, 2, ·· · , n. In this case, the isometry subdivision of input space Δ (a) is also called m- mesh subdivision. Under this convention, it is not difficult to obtain the generalized cube Δ (a) which can be decomposed into mⁿ generalized small polyhedrons Δ_j with right angle side length as $\frac{a}{m}$ , and $Δ (a) = \cup_{j = 1}^{m^{n}} Δ_{j}$ .

Construction of a piecewise linear function S: According to Refs. [26, 27], the input space Δ (a) is divided into m- mesh. Let Δ_{i₁i₂···i_n} be a small n- dimensional polyhedron after subdivision, and $x_{k}^{*} = (x_{1}^{k}, x_{2}^{k}, \cdot \cdot \cdot, x_{n}^{k})$ is denoted as the k-th vertex coordinate of n- dimensional small polyhedron Δ_{i₁i₂···i_n} after subdivision, and suppose that these n + 1 vertices are not on the same hyperplane, so as to ensure that the following determinant |D_n| ≠ 0. If n + 1 vertices coordinates of Δ_{i₁i₂···i_n} are briefly note as $x_{1}^{*}, x_{2}^{*}, \cdot \cdot \cdot, x_{n}^{*}, x_{n + 1}^{*}$ in the specified order, then each $f (x_{k}^{*})$ can take the corresponding value under the action of f, and the vertices coordinates of each small polyhedron Δ_{i₁i₂···i_n} in $ℝ^{n + 1}$ can be expressed as follows:

$\begin{matrix} (x_{1}^{1}, x_{2}^{1}, \cdot \cdot \cdot, x_{n}^{1}, f (x_{1}^{*})), (x_{1}^{2}, x_{2}^{2}, \cdot \cdot \cdot, x_{n}^{2}, f (x_{2}^{*})), \\ \cdot \cdot \cdot, (x_{1}^{n + 1}, x_{2}^{n + 1}, \cdot \cdot \cdot, x_{n}^{n + 1}, f (x_{n + 1}^{*})) . \end{matrix}$ (1)

Let the hyperplane S be determined by n + 1 vertices of form (1) of Δ_{i₁i₂···i_n} in $ℝ^{n + 1}$ , and the analytic expression of the linear function S is expressed as

$\begin{matrix} S_{i_{1} i_{2} \cdot \cdot \cdot i_{n}} (x) \\ = b_{i_{1}}^{1} x_{1} + b_{i_{2}}^{2} x_{2} + \cdot \cdot \cdot + b_{i_{n}}^{n} x_{n} + λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{}, \end{matrix}$ (2) where x = (x₁, x₂, ·· · , x_n) ∈ Δ_{i₁i₂···i_n}, $i_{j} = 1, 2, \cdot \cdot \cdot, m$ ; j = 1, 2, ·· · , n. In addition, because the hyperplane S_{i₁i₂···i_n} (x) is formed by cutting surface f (x) in the sense of m- mesh, and n + 1 vertices $x_{k}^{*}$ on $ℝ^{n + 1}$ are the common intersection of hyperplane S_{i₁i₂···i_n} (x) and surface f (x), it is necessary for each vertex $x_{k}^{*}$ coordinate to have $S_{i_{1} i_{2} \cdot \cdot \cdot i_{n}} (x_{k}^{*}) = f (x_{k}^{*})$ , k = 1, 2, ·· · , n, n + 1.

By substituting the coordinates of n + 1 vertices of Δ_{i₁i₂···i_n} into formula (2), a set of hyperplane linear equations on $ℝ^{n + 1}$ can be obtained as follows:

${\begin{matrix} b_{i_{1}}^{1} x_{1}^{1} + b_{i_{2}}^{2} x_{2}^{1} + \cdot \cdot \cdot + b_{i_{n}}^{n} x_{n}^{1} + λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{} = f (x_{1}^{*}) \\ b_{i_{1}}^{1} x_{1}^{2} + b_{i_{2}}^{2} x_{2}^{2} + \cdot \cdot \cdot + b_{i_{n}}^{n} x_{n}^{2} + λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{} = f (x_{2}^{*}) \\ \dots \dots \dots \dots \dots \dots \dots \\ b_{i_{1}}^{1} x_{1}^{n} + b_{i_{2}}^{2} x_{2}^{n} + \cdot \cdot \cdot + b_{i_{n}}^{n} x_{n}^{n} + λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{} = f (x_{n}^{*}) \\ b_{i_{1}}^{1} x_{1}^{n + 1} + b_{i_{2}}^{2} x_{2}^{n + 1} + \cdot \cdot \cdot + b_{i_{n}}^{n} x_{n}^{n + 1} + λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{} = f (x_{n + 1}^{*}) \end{matrix} .$ (3)

By substituting the coordinates of n + 1 vertices of Δ_{i₁i₂···i_n} into formula (2), a set of hyperplane linear equations on $ℝ^{n + 1}$ can be obtained as follows:

Because all vertex coordinates of each small polyhedron Δ_{i₁i₂···i_n} in $ℝ^{n + 1}$ shown in formula (1) are known in the sense of m- mesh subdivision. Therefore, all coefficients ${b_{i_{1}}^{1}, b_{i_{2}}^{2}, \cdot \cdot \cdot, b_{i_{n}}^{n}, λ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{}}$ in equation group (3) can be regarded as unknown quantities, and then the values of these coefficients can be obtained by solving equation group (3). According to Ref. [27], the analytic expression of the piecewise linear function S on Δ (a) is

$\begin{matrix} S (x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \\ = {\begin{matrix} \frac{| D_{i_{1}}^{1} |}{| D_{n} |} x_{1} + \frac{| D_{i_{2}}^{2} |}{| D_{n} |} x_{2} + \cdot \cdot \cdot + \frac{| D_{i_{n}}^{n} |}{| D_{n} |} x_{n} + \frac{| D_{i_{n + 1}}^{n + 1} |}{| D_{n} |}, \\ (x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \in Δ_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}, i_{1}, i_{2} \cdot \cdot \cdot, i_{n} \in {1, 2, \cdot \cdot \cdot, m} \\ 0, Otherwise \end{matrix} . \end{matrix}$ (4)

Here, the coefficients $| D_{j}^{i_{j}} | / - | D_{n} |$ (j = 1, 2, ·· · , n, n + 1) of the piecewise linear function S on each hyperplane Δ_{i₁i₂···i_n} are given in the following matrix determinant form, i.e.,

$\begin{matrix} \begin{matrix} | D_{i_{1}}^{1} | = | \begin{matrix} f (x_{1}^{*}) & x_{2}^{1} & \dots & x_{n}^{1} & 1 \\ f (x_{2}^{*}) & x_{2}^{2} & \dots & x_{n}^{2} & 1 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ f (x_{n}^{*}) & x_{2}^{n} & \dots & x_{n}^{n} & 1 \\ f (x_{n + 1}^{*}) & x_{2}^{n + 1} & \dots & x_{n}^{n + 1} & 1 \end{matrix} |, \end{matrix} \end{matrix}$

$\begin{matrix} | D_{i_{2}}^{2} | = | \begin{matrix} x_{1}^{1} & f (x_{1}^{*}) & \dots & x_{n}^{1} & 1 \\ x_{1}^{2} & f (x_{2}^{*}) & \dots & x_{n}^{2} & 1 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{1}^{n} & f (x_{n}^{*}) & \dots & x_{n}^{n} & 1 \\ x_{1}^{n + 1} & f (x_{n + 1}^{*}) & \dots & x_{n}^{n + 1} & 1 \end{matrix} |, \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots, \\ | D_{i_{n}}^{n} | = | \begin{matrix} x_{1}^{1} & x_{2}^{1} & \dots & f (x_{1}^{*}) & 1 \\ x_{1}^{2} & x_{2}^{2} & \dots & f (x_{2}^{*}) & 1 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{1}^{n} & x_{2}^{n} & \dots & f (x_{n}^{*}) & 1 \\ x_{1}^{n + 1} & x_{2}^{n + 1} & \dots & f (x_{n + 1}^{*}) & 1 \end{matrix} |, \\ | D_{n} | = | \begin{matrix} x_{1}^{1} & x_{2}^{1} & \dots & x_{n}^{1} & 1 \\ x_{1}^{2} & x_{2}^{2} & \dots & x_{n}^{2} & 1 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ x_{1}^{n} & x_{2}^{n} & \dots & x_{n}^{n} & 1 \\ x_{1}^{n + 1} & x_{2}^{n + 1} & \dots & x_{n}^{n + 1} & 1 \end{matrix} | . \end{matrix}$ (5)

Moreover, by applying m- mesh subdivision, it is not difficult to know that the vertex coordinates of each small polyhedron Δ_{i₁i₂···i_n} can be written as $x_{k}^{*} = (\frac{a i_{1}}{m}, \frac{a i_{2}}{m}, \cdot \cdot \cdot, \frac{a i_{n}}{m})$ , and the index of each coordinate axis is i₁, i₂, ·· · , i_n ∈ {1, 2, ·· · , m}. See [27, 30].

In fact, with the specific analytic expression (4) of piecewise linear function (PLF), people naturally wonder whether PLFs can approximate a multivariate continuous function with any precision? Only by admitting this fact can we continue to study the problem of Mamdani fuzzy system approximating PLF or even continuous function, in which PLF is only a bridge in the process of proof. On this question, Ref. [27] gives a positive answer. Unfortunately, the authors only guessed that the sum $(1 + a \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | / - m | | D_{n} | |)$ (called approximation factor) is a constant in [27], and did not give a detailed proof. However, this problem will hinder Mamdani fuzzy system to approach a continuous function in theory. Therefore, we prove that the approximation factor in [27] is indeed a constant through the properties of the coefficient determinants and matrix norm of PLF formula (4), which has important theoretical significance.

3 Approximation of piecewise linear functions

Actually, Mamdani fuzzy system is one of the simplest fuzzy system models. Its main feature is that each rule’s subsequent output is a fuzzy set, while the subsequent output of T-S fuzzy system is a multivariate linear function about input variables. Next, we first review some related knowledge of Mamdani fuzzy system, and assume that the rule base is composed of the following fuzzy rules

$\begin{matrix} R_{i_{1} i_{2} \dots i_{n}} : If x_{1} is A_{i_{1}}^{1}, x_{2} is A_{i_{2}}^{2}, \dots, \\ x_{n} is A_{i_{n}}^{n}, then u is C_{i_{1} i_{2} \dots i_{n}} . \end{matrix}$

The input-output relationship of Mamdani fuzzy system with single point fuzzification, product reasoning machine and central average fuzzification is as follows:

$\begin{matrix} F (x_{1}, x_{2}, \cdot \cdot \cdot, x_{n}) \\ = \frac{\sum_{i_{1} = 1}^{N_{1}} \sum_{i_{2} = 1}^{N_{2}} \cdot \cdot \cdot \sum_{i_{n} = 1}^{N_{n}} (\prod_{k = 1}^{n} A_{i_{k}}^{k} (x_{k})) \cdot {\bar{y}}_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}}{\sum_{i_{1} = 1}^{N_{1}} \sum_{i_{2} = 1}^{N_{2}} \cdot \cdot \cdot \sum_{i_{n} = 1}^{N_{n}} (\prod_{k = 1}^{n} A_{i_{k}}^{k} (x_{j}))}, \end{matrix}$ (6) where $x = (x_{1}, x_{2}, \dots, x_{n}) \in U_{1} \times U_{2} \times \dots U_{n} \subset ℝ^{n}$ is an input variable, ${A_{i_{1}}^{1}, A_{i_{2}}^{2}, \dots, A_{i_{n}}^{n}}$ is a family of antecedent fuzzy sets of Mamdani fuzzy system corresponding to the i_j-th coordinate axis, i.e., $A_{i_{j}}^{j} \in F (U_{j})$ , j = 1, 2, ⋯ , n, i_j = 1, 2, ⋯ , N_j, and C_{i₁i₂⋯i_n} is a consequent fuzzy set corresponding to rule R_{i₁i₂⋯i_n} in the output domain $V \subset ℝ$ , the ${\bar{y}}^{i_{1} i_{2} \cdot \cdot \cdot i_{n}}$ is the center of the consequent fuzzy set C_{i₁i₂⋯i_n}, the real number u is the output variable on V. See [27].

In addition, it is not hard to see that the total number of all possible fuzzy rules in Mamdani fuzzy system is M = N₁ × N₂ × ·· · × N_n, and ${A_{1}^{1}, A_{2}^{1}, \cdot \cdot \cdot, A_{N_{1}}^{1}}, {A_{1}^{2}, A_{2}^{2}, \cdot \cdot \cdot, A_{N_{2}}^{2}}$ , $\cdot \cdot \cdot, {A_{1}^{n}, A_{2}^{n}, \cdot \cdot \cdot, A_{N_{n}}^{n}}$ are the antecedent fuzzy sets on the i_j-th coordinate axis, respectively. For simplicity, N₁ = N₂ = ⋯ = N_n = m is chosen in this paper. See [27, 30].

Lemma 1. ^[27] Let S be a piecewise linear function of form (4) on the generalized cube Δ (a), where a > 0, then, for arbitrary ɛ > 0, there is a $m_{0} \in ℕ$ such that Mamdani fuzzy system F_m of form (6) determined by the number of subdivisions m satisfies || F_m - S ||_∞ < ɛ when m > m₀.

In accordance with the properties of determinant, it is not difficult to rewrite the above matrix determinant (5) as follows:

$\begin{matrix} | D_{i_{1}}^{1} | = | \begin{matrix} f (x_{1}^{*}) - f (x_{2}^{*}) & x_{2}^{1} - x_{2}^{2} & \cdot \cdot \cdot & x_{n}^{1} - x_{n}^{2} \\ f (x_{2}^{*}) - f (x_{3}^{*}) & x_{2}^{2} - x_{2}^{3} & \cdot \cdot \cdot & x_{n}^{2} - x_{n}^{3} \\ ⋮ & ⋮ & ⋮ \\ f (x_{n}^{*}) - f (x_{n + 1}^{*}) & x_{2}^{n} - x_{2}^{n + 1} & \cdot \cdot \cdot & x_{n}^{n} - x_{n}^{n + 1} \end{matrix} |, \\ | D_{i_{2}}^{2} | = | \begin{matrix} x_{1}^{1} - x_{1}^{2} & f (x_{1}^{*}) - f (x_{2}^{*}) & \cdot \cdot \cdot & x_{n}^{1} - x_{n}^{2} \\ x_{1}^{2} - x_{1}^{3} & f (x_{2}^{*}) - f (x_{3}^{*}) & \cdot \cdot \cdot & x_{n}^{2} - x_{n}^{3} \\ ⋮ & ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & f (x_{n}^{*}) - f (x_{n + 1}^{*}) & \cdot \cdot \cdot & x_{n}^{n} - x_{n}^{n + 1} \end{matrix} |, \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots, \\ | D_{i_{n}}^{n} | = | \begin{matrix} x_{1}^{1} - x_{1}^{2} & x_{2}^{1} - x_{2}^{2} & \cdot \cdot \cdot & f (x_{1}^{*}) - f (x_{2}^{*}) \\ x_{1}^{2} - x_{1}^{3} & x_{2}^{2} - x_{2}^{3} & \cdot \cdot \cdot & f (x_{2}^{*}) - f (x_{3}^{*}) \\ ⋮ & ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & x_{2}^{n} - x_{2}^{n + 1} & \cdot \cdot \cdot & f (x_{n}^{*}) - f (x_{n + 1}^{*}) \end{matrix} |, \\ | D_{n} | = | \begin{matrix} x_{1}^{1} - x_{1}^{2} & x_{2}^{1} - x_{2}^{2} & \cdot \cdot \cdot & x_{n}^{1} - x_{n}^{2} \\ x_{1}^{2} - x_{1}^{3} & x_{2}^{2} - x_{2}^{3} & \cdot \cdot \cdot & x_{n}^{2} - x_{n}^{3} \\ ⋮ & ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & x_{2}^{n} - x_{2}^{n + 1} & \cdot \cdot \cdot & x_{n}^{n} - x_{n}^{n + 1} \end{matrix} | . \end{matrix}$ (7)

Note 2. According to the m- mesh subdivision, the difference of all adjacent coordinate components in these determinants on the same coordinate axis $x_{i}^{j} - x_{i}^{j + 1}$ can only be $\pm \frac{a}{m}$ or zero.

If these determinants (except f) are expanded in column 1, column 2,. . . , column n, and then n algebraic cofactors of n - 1 order m can be obtained by combining and sorting out the factors $(f (x_{i}^{*}) - f (x_{i + 1}^{*}))$ item by item, where the n - 1 order algebraic cofactors corresponding to the factor $(f (x_{1}^{*}) - f (x_{2}^{*}))$ are simply expressed as $| B_{1}^{1} |, | B_{2}^{1} |, \cdot \cdot \cdot, | B_{n}^{1} |;$ the factor $(f (x_{2}^{*}) - f (x_{3}^{*}))$ corresponds to $| B_{1}^{2} |, | B_{2}^{2} |, \cdot \cdot \cdot, | B_{n}^{2} |$ . By analogy, the factor $(f (x_{i}^{*}) - f (x_{i + 1}^{*}))$ corresponds to the n - 1 order algebraic cofactors are $| B_{1}^{i} |, | B_{2}^{i} |, \cdot \cdot \cdot, | B_{k}^{i} |, \cdot \cdot \cdot, | B_{n}^{i} |$ , where $| B_{k}^{i} |$ represents the algebraic cofactor of the i-th determinant, the column k and the i-th element. For example, $| B_{1}^{1} |$ is a cofactor of $| D_{i_{1}}^{1} |$ , $| B_{2}^{1} |$ and $| B_{2}^{2} |$ are cofactors of $| D_{i_{2}}^{1} |$ , $| B_{n}^{2} |$ is a cofactor of $| D_{i_{n}}^{n} |$ , and

$\begin{matrix} \begin{matrix} | B_{1}^{1} | = (- 1)^{2} | \begin{matrix} x_{2}^{2} - x_{2}^{3} & \cdot \cdot \cdot & x_{n}^{2} - x_{n}^{3} \\ ⋮ & ⋮ \\ x_{2}^{n} - x_{2}^{n + 1} & \cdot \cdot \cdot & x_{n}^{n} - x_{n}^{n + 1} \end{matrix} |, \\ | B_{2}^{1} | = (- 1)^{3} | \begin{matrix} x_{1}^{2} - x_{1}^{3} & \cdot \cdot \cdot & x_{n}^{2} - x_{n}^{3} \\ ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & \cdot \cdot \cdot & x_{1}^{n} - x_{1}^{n + 1} \end{matrix} | \\ | B_{2}^{2} | = (- 1)^{4} | \begin{matrix} x_{1}^{1} - x_{1}^{2} & \cdot \cdot \cdot & x_{n}^{1} - x_{n}^{2} \\ ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & \cdot \cdot \cdot & x_{n}^{n} - x_{n}^{n + 1} \end{matrix} |, \\ | B_{n}^{2} | = (- 1)^{n + 2} | \begin{matrix} x_{1}^{1} - x_{1}^{2} & \cdot \cdot \cdot & x_{n - 1}^{1} - x_{n - 1}^{2} \\ ⋮ & ⋮ \\ x_{1}^{n} - x_{1}^{n + 1} & \cdot \cdot \cdot & x_{n - 1}^{n} - x_{n - 1}^{n + 1} \end{matrix} | \end{matrix}, \end{matrix}$

It should be noted that in the sense of isometric subdivision, the vertex coordinates of each small polyhedron Δ_{i₁i₂···i_n} as $x_{k}^{*} = (\frac{a i_{1}}{m}, \frac{a i_{2}}{m}, \cdot \cdot \cdot, \frac{a i_{n}}{m})$ . Hence, the difference $x_{i}^{j} - x_{i}^{j + 1}$ of all adjacent coordinate components on the same coordinate axis in these algebraic cofactor is only $\pm \frac{a}{m}$ or zero.

Definition 3.1. Let matrix A be n square array, let ||A|| = | (| A| ) |, then ||A|| is called matrix norm of A, that is, matrix norm ||A|| is absolute value of determinant of A. Obviously, the matrix norm of any square array A always satisfies ||A||≥0.

Lemma 2. ^[27] Let f be continuous function on compact set $Δ (a) \subset ℝ^{n}$ , (x ; f (x) ) is a given data pair, but the analytic expression of f is unknown. Then, for any ɛ > 0, there is a subdivision number $m \in ℕ$ and the piecewise linear function S of form (4), which satisfies the requirements in the sense of infinite norm ∥S - f∥ _∞ < $(\frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | + 1) ɛ$ , where the infinite norm is defined as ${∥ S - f ∥}_{\infty} = sup_{x \in Δ (a)} | S (x) - f (x) |$ .

It is not difficult to see that Lemma 1 and Lemma 2 can be utilized to obtain the following Lemma 3. That is to say, Mamdani fuzzy system F_m of form (6) can indeed approximate f to arbitrary accuracy with respect to infinite norm.

Lemma 3. ^[27] If f be a continuous function on compact set $Δ (a) \subset ℝ^{n}$ , then, for arbitrary ɛ > 0, there is the number of subdivisions $m_{0} \in ℕ$ and the Mamdani fuzzy system of form (6), such that || F_m - f ||_∞ < ɛ when m > m₀, that is, F_m can approximate f to any precision by infinite norm.

However, it must be said that it is a pity, because the Ref. [27] does not give strict proof in theory that the sum factor $(\frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | + 1)$ in Lemma 2 is a constant independent of the number of subdivisions m, but simply expounds it in language. Thus, in this paper we will prove that the sum factor is a constant independent of the number of subdivision m on $ℝ^{n}$ .

Definition 3.2. Assume that matrix $| B_{k}^{i} |$ (i, k = 1, 2, ·· · , n) are the corresponding n - 1 order algebraic cofactors as described above, and $| | B_{k}^{i} | |$ and ||D_n|| are matrix norms, then the expression $(\frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | + 1)$ is called the approximation factor of ∥f - S ∥ _∞, also referred to as the approximation factor of a piecewise linear function S.

In fact, the expression of the approximation factor is more complex with the increasing dimension of the input variable n. Moreover, the isometric subdivision of the generalized cube Δ (a) no longer exists when n ≥ 4, which makes it difficult to find the n - 1 order algebraic cofactors.

4 Approximation factor and subdivision

According to Lemma 2, only if the approximation factor is a constant independent of the number of subdivision, the piecewise linear function S has the approximation, so it is very important whether the approximation factor is a constant independent of the number of subdivision. This conclusion is only a conjecture in [27], but has not been proved in detail. Therefore, in this paper we will prove that the approximation factor is indeed a constant independent of the number of subdivision m on $ℝ^{n}$ .

In order to verify whether the approximation factor is constant, it is necessary to determin the vertex coordinates of each small polyhedron Δ_{i₁i₂···i_n} in $ℝ^{n}$ and its order, and then find the n - 1 order algebraic cofactor and its matrix norm $| | B_{k}^{i} | |$ corresponding to factor $(f (x_{i}^{*}) - f (x_{i + 1}^{*}))$ according to Note 2. Next, take n = 3 as an example to continue to explore the vertex coordinates of small tetrahedron Δ_{i₁i₂···i_n} on $ℝ^{n}$ , and its sorting problem. See Fig. 1.

Fig. 1

Subdivision image of a small cube with a side length of $\frac{a}{m}$ when n = 3.

In fact, for any input variable (x₁, x₂, x₃) ∈ [0, a] ³, there are the indexes i₁, i₂, i₃ $\in ℕ$ , such that (x₁, x₂, x₃) ∈ Δ_{i
₁
i
₂
i
₃}. Suppose all coordinates of four vertices of the small tetrahedron Δ_{i
₁
i
₂
i
₃} are $x_{1}^{*} = (\frac{(i_{1} + 1) a}{m}, \frac{(i_{2} + 1) a}{m}, \frac{i_{3} a}{m})$ , $x_{2}^{*} = (\frac{i_{1} a}{m}, \frac{(i_{2} + 1) a}{m}, \frac{i_{3} a}{m})$ , $x_{3}^{*} = (\frac{(i_{1} + 1) a}{m}, \frac{i_{2} a}{m}, \frac{i_{3} a}{m})$ and $x_{4}^{*} = (\frac{(i_{1} + 1) a}{m}, \frac{(i_{2} + 1) a}{m}, \frac{(i_{3} + 1) a}{m})$ respectively, as shown in Fig. 1.

Note 3. It is not difficult to find from Fig. 1 that if $x_{1}^{*}$ is taken as the base point, the order of vertices $x_{1}^{*}$ and $x_{2}^{*}, x_{3}^{*}, x_{4}^{*}$ conforms to the right-hand rule, that is, $x_{1}^{*}$ is taken as the base point, and the bending direction of the four fingers of the right hand is from $\vec{x_{1}^{*} x_{2}^{*}}$ to $\vec{x_{2}^{*} x_{3}^{*}}$ , so the order of $x_{2}^{*}, x_{3}^{*}, x_{4}^{*}$ is determined in turn. According to this sort, we find that the corresponding coordinates have the following rules: the straight-line distance of vertices $x_{1}^{*}$ and $x_{2}^{*}, x_{3}^{*}, x_{4}^{*}$ along their respective coordinate axes is $\frac{a}{m}$ , where the difference between the $x_{2}^{*}$ -first component, $x_{3}^{*}$ -second component and $x_{4}^{*}$ -third component of vertices and the $x_{1}^{*}$ -corresponding coordinate component are all $\frac{a}{m}$ , while the other coordinate components have no change (difference is zero), that is, the difference between the components of adjacent vertices on the same coordinate axis is $\pm \frac{a}{m}$ or zero.

Similarly, the method can be extended to n- dimensional polyhedron. Specific method: firstly, select a vertex on n- dimensional polyhedron Δ_{i₁i₂···i_n} as the base point $x_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{*}$ , let the base point be $x_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{*} = (\frac{i_{1} a}{m}, \frac{i_{2} a}{m}, \cdot \cdot \cdot, \frac{i_{n - 1} a}{m}, \frac{i_{n} a}{m})$ , and then rotate each coordinate component $\frac{i_{j} a}{m}$ to $\frac{(i_{j} - 1) a}{m}$ or $\frac{(i_{j} + 1) a}{m}$ (j = 1, 2, ·· · , n) along its coordinate axis i_j in turn. If the other components are invariant, then the total n + 1 vertex coordinates of Δ_{i₁i₂···i_n} can be obtained.

Thorem 4.1. Let f be a continuous function on compact set $Δ (a) \subset ℝ^{n}$ , the Δ_{i₁i₂···i_n} be the small polyhedron of the above n-dimensional subdivision, and the coordinates of all vertices are denoted as $x_{i_{1} i_{2} \cdot \cdot \cdot i_{n}}^{*} = (\frac{i_{1} a}{m}, \frac{i_{2} a}{m}, \cdot \cdot \cdot, \frac{i_{n} a}{m})$ , i₁, i₂, ·· · , i_n ∈ {1, 2, ·· · , m}. Matrix determinant $| B_{k}^{i} |$ is the algebraic cofactor of n - 1 order defined in Note 2, i, k = 1, 2, ·· · , n, $| | B_{k}^{i} | |$ and ||D_n|| are matrix norms, then the approximation factor $(\frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | + 1)$ is independent of the number of subdivision m.

Proof. According to Ref. [25], for any input variable (x₁, x₂, ·· · , x_n) ∈ Δ (a), there is m-mesh subdivision and index i₁, i₂, ·· · , i_n ∈ {1, 2, ·· · , m} on cube Δ (a), such that (x₁, x₂, ·· · , x_n) ∈ Δ_{i₁i₂···i_n}, where the right angle side length of each small polyhedron Δ_{i₁i₂···i_n} is $\frac{a}{m}$ .

Without losing generality, we can choose vertex $(\frac{(1 + i_{1}) a}{m}, \frac{(1 + i_{2}) a}{m}, \frac{(1 + i_{3}) a}{m}, \cdot \cdot \cdot, \frac{(1 + i_{n - 1}) a}{m}, \frac{i_{n} a}{m})$ as the base point of Δ_{i₁i₂···i_n}, the other n vertex coordinates are $\begin{matrix} \begin{matrix} (\frac{i_{1} a}{m}, \frac{(1 + i_{2}) a}{m}, \frac{(1 + i_{3}) a}{m}, \cdot \cdot \cdot, \frac{(1 + i_{n - 1}) a}{m}, \frac{i_{n} a}{m}), \\ (\frac{(1 + i_{1}) a}{m}, \frac{i_{2} a}{m}, \frac{(1 + i_{3}) a}{m}, \cdot \cdot \cdot, \frac{(1 + i_{i - 1}) a}{m}, \frac{i_{n} a}{m}), \\ (\frac{(1 + i_{1}) a}{m}, \frac{(1 + i_{2}) a}{m}, \frac{i_{3} a}{m}, \cdot \cdot \cdot, \frac{(1 + i_{n - 1}) a}{m}, \frac{i_{n} a}{m}), \dots, \\ (\frac{(1 + i_{1}) a}{m}, \frac{(1 + i_{2}) a}{m}, \frac{(1 + i_{3}) a}{m}, \cdot \cdot \cdot, \frac{(1 + i_{n - 1}) a}{m}, \frac{(1 + i_{n}) a}{m}) . \end{matrix} \end{matrix}$

By Note 3, these vertex coordinates are directly substituted into the formula (7) to obtain $\begin{matrix} | D_{n} | & = | \begin{matrix} \frac{a}{m} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \cdot \cdot \cdot & \frac{a}{m} & 0 \\ 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | \\ = - (\frac{a}{m})^{n} \Rightarrow ∥ D_{n} ∥ = (\frac{a}{m})^{n} \end{matrix}$

According to Note 2, the difference $x_{i}^{j} - x_{i}^{j + 1}$ of adjacent coordinate components on the same coordinate axis in matrix determinant $| D_{i_{1}}^{1} |$ , $| D_{i_{2}}^{2} |$ , ···, $| D_{i_{n}}^{n} |$ can only be $\pm \frac{a}{m}$ or zero. If the above vertex coordinates are directly substituted into formula (7), it can be obtained immediately.

$\begin{matrix} \begin{matrix} | D_{i_{1}}^{1} | = | \begin{matrix} f (x_{1}^{*}) - f (x_{2}^{*}) & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ f (x_{2}^{*}) - f (x_{3}^{*}) & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ f (x_{3}^{*}) - f (x_{4}^{*}) & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ f (x_{4}^{*}) - f (x_{5}^{*}) & 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ f (x_{n}^{*}) - f (x_{n + 1}^{*}) & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} |, \\ | D_{i_{2}}^{2} | = | \begin{matrix} \frac{a}{m} & f (x_{1}^{*}) - f (x_{2}^{*}) & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & f (x_{2}^{*}) - f (x_{3}^{*}) & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & f (x_{3}^{*}) - f (x_{4}^{*}) & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & f (x_{4}^{*}) - f (x_{5}^{*}) & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & f (x_{n}^{*}) - f (x_{n + 1}^{*}) & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} |, \end{matrix} \end{matrix}$ $\begin{matrix} \begin{matrix} | D_{i_{3}}^{3} | = | \begin{matrix} \frac{a}{m} & 0 & f (x_{1}^{*}) - f (x_{2}^{*}) & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & f (x_{2}^{*}) - f (x_{3}^{*}) & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & f (x_{3}^{*}) - f (x_{4}^{*}) & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & f (x_{4}^{*}) - f (x_{5}^{*}) & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & f (x_{n}^{*}) - f (x_{n + 1}^{*}) & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} |, \\ \dots \dots, \\ | D_{i_{n}}^{n} | = | \begin{matrix} \frac{a}{m} & 0 & 0 & \cdot \cdot \cdot & 0 & f (x_{1}^{*}) - f (x_{2}^{*}) \\ - \frac{a}{m} & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & f (x_{2}^{*}) - f (x_{3}^{*}) \\ 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & f (x_{3}^{*}) - f (x_{4}^{*}) \\ 0 & 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & f (x_{4}^{*}) - f (x_{5}^{*}) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & f (x_{n}^{*}) - f (x_{n + 1}^{*}) \end{matrix} | . \end{matrix} \end{matrix}$

By Note 2, the n - 1 order algebraic cofactors are obtained by expanding $| D_{i_{1}}^{1} |$ with column 1, i.e., $\begin{matrix} \begin{matrix} | B_{1}^{1} | = | \begin{matrix} \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = - (\frac{a}{m})^{n - 1}, \\ | B_{1}^{2} | = | \begin{matrix} 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = 0, \\ | B_{1}^{3} | = | \begin{matrix} 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = 0, \cdot \cdot \cdot, \\ | B_{1}^{n} | = | \begin{matrix} 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ \frac{a}{m} & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & \frac{a}{m} & 0 \end{matrix} | = 0 . \end{matrix} \end{matrix}$

In the same way, the n - 1 order algebraic cofactors are obtained by expanding $| D_{i_{2}}^{2} |$ with column 2, that is, $\begin{matrix} \begin{matrix} | B_{2}^{1} | = | \begin{matrix} - \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = (\frac{a}{m})^{n - 1}, \\ | B_{2}^{2} | = | \begin{matrix} \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = - (\frac{a}{m})^{n - 1}, \\ | B_{2}^{3} | = | \begin{matrix} \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & - \frac{a}{m} \end{matrix} | = 0 \dots, \\ | B_{2}^{n} | = | \begin{matrix} \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & \cdot \cdot \cdot & \frac{a}{m} & 0 \end{matrix} | = 0 . \end{matrix} \end{matrix}$

Analogously, the determinant $| D_{i_{n}}^{n} |$ is expanded by column n to obtain n - 1 order algebraic cofactors as follows: $\begin{matrix} \begin{matrix} | B_{n}^{1} | = | \begin{matrix} - \frac{a}{m} & \frac{a}{m} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & \frac{a}{m} \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & - \frac{a}{m} \end{matrix} |, \end{matrix} \end{matrix} \begin{matrix} = (- 1)^{n - 1} (\frac{a}{m})^{n - 1}, \end{matrix}$ $\begin{matrix} \begin{matrix} | B_{n}^{2} | = | \begin{matrix} \frac{a}{m} & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & \frac{a}{m} \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & - \frac{a}{m} \end{matrix} | \\ = (- 1)^{n - 2} (\frac{a}{m})^{n - 1}, \\ | B_{n}^{3} | = | \begin{matrix} \frac{a}{m} & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & 0 & - \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & \frac{a}{m} \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & - \frac{a}{m} \end{matrix} |, \dots \dots, \\ = (- 1)^{n - 3} (\frac{a}{m})^{n - 1}, \\ | B_{n}^{n} | = | \begin{matrix} \frac{a}{m} & 0 & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ - \frac{a}{m} & \frac{a}{m} & 0 & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & - \frac{a}{m} & \frac{a}{m} & 0 & \cdot \cdot \cdot & 0 & 0 \\ 0 & 0 & - \frac{a}{m} & \frac{a}{m} & \cdot \cdot \cdot & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & \frac{a}{m} & 0 \\ 0 & 0 & 0 & 0 & \cdot \cdot \cdot & - \frac{a}{m} & \frac{a}{m} \end{matrix} | \\ = (\frac{a}{m})^{n - 1} . \end{matrix} \end{matrix}$

In summary, after the determinant $| D_{i_{1}}^{1} |$ is expanded according to the column 1, except for the n - 1 order algebraic cofactor $| B_{1}^{1} | = - (\frac{a}{m})^{n - 1}$ , all other n - 1 order algebraic cofactors are zero, that is, $| B_{1}^{i} | = 0$ , i = 2, 3, ·· · , n; the $| D_{i_{2}}^{2} |$ is expanded according to the column 2, except for the n - 1 order algebraic cofactors $| B_{2}^{1} | = (\frac{a}{m})^{n - 1}$ and $| B_{2}^{2} | = - (\frac{a}{m})^{n - 1}$ , all other n - 1 order algebraic cofactors are zero, i.e., $| B_{2}^{i} | = 0$ , i = 3, 4, ·· · , n. Similarly, the $| D_{i_{3}}^{3} |$ is expanded as column 3, except for $| B_{3}^{1} | = - (\frac{a}{m})^{n - 1}$ , $| B_{3}^{3} | = - (\frac{a}{m})^{n - 1}$ and $| B_{3}^{2} | = (\frac{a}{m})^{n - 1}$ , all others have $| B_{3}^{4} | = | B_{3}^{5} | = \cdot \cdot \cdot = | B_{3}^{n} | = 0$ . Generally, the determinant $| D_{i_{n}}^{n} |$ is expanded by column n, we always have $\begin{matrix} | B_{n}^{1} | = (- 1)^{n - 1} (\frac{a}{m})^{n - 1}, | B_{n}^{2} | = (- 1)^{n - 2} (\frac{a}{m})^{n - 1}, \\ | B_{n}^{3} | = (- 1)^{n - 3} (\frac{a}{m})^{n - 1}, \dots, | B_{n}^{n} | = (\frac{a}{m})^{n - 1} . \end{matrix}$

Finally, by Definition 3.2 we can immediately get that $\begin{matrix} \begin{matrix} \frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | = \frac{a}{m | | D_{n} | |} \\ [(| | B_{1}^{1} | | + | | B_{1}^{2} | | + \cdot \cdot \cdot + | | B_{1}^{n} | |) \\ + (| | B_{2}^{1} | | + | | B_{2}^{2} | | + \cdot \cdot \cdot + | | B_{2}^{n} | |) + \\ (| | B_{3}^{1} | | + | | B_{3}^{2} | | + \cdot \cdot \cdot + | | B_{3}^{n} | |) + \cdot \cdot \cdot \\ + (| | B_{n}^{1} | | + | | B_{n}^{2} | | + \cdot \cdot \cdot + | | B_{n}^{n} | |)] \\ = \frac{a}{m | | D_{n} | |} [(\frac{a}{m})^{n - 1} + 2 (\frac{a}{m})^{n - 1} + 3 (\frac{a}{m})^{n - 1} \\ + \cdot \cdot \cdot + n (\frac{a}{m})^{n - 1}] \\ = \frac{a}{m (a / - m)^{n}} \frac{n (n + 1)}{2} (\frac{a}{m})^{n - 1} \\ = \frac{n (n + 1)}{2} . \end{matrix} \end{matrix}$

Hence, the approximation factor is $(\frac{a}{m | | D_{n} | |} \sum_{i = 1}^{n} \sum_{k = 1}^{n} | | B_{k}^{i} | | + 1) = \frac{n (n + 1) + 2}{2}$ . Clearly, It is only related to the dimension n of input space, but has nothing to do with the number of subdivision m.

Actually, although the number of subdivision m of input space Δ (a) is independent of the approximation factor, the number of subdivision m is closely related to the corresponding PLFs, Mamdani fuzzy system and approximation accuracy. For example, PLFs are mainly constructed by the subdivision of input space. Generally speaking, the larger the m value, the finer the subdivision, the better the approximation accuracy, but the greater the complexity. Conversely, if the m value is too small, although the complexity is reduced, it may not achieve the required approximation accuracy. So far, the problems left by [27] have been solved.

In addition, from the construction of Mamdani fuzzy system (6), the antecedent fuzzy sets ${A_{1}^{1}, A_{2}^{1}, \cdot \cdot \cdot, A_{N_{1}}^{1}}, {A_{1}^{2}, A_{2}^{2}, \cdot \cdot \cdot, A_{N_{2}}^{2}}$ and ${A_{1}^{3}, A_{2}^{3}, \cdot \cdot \cdot, A_{N_{3}}^{3}}$ on the three coordinate axes also depend on the number of subdivisions m, but in this paper we assume that there is always N₁ = N₂ = N₃ = m.

5 Example analysis

In this part, the important role of approximation factor of a piecewise linear function S is illustrated through a practical example. For simplicity, we may extend the condition of the given binary pair ((x, y) ; 1ptf (x, y)) to the analytic expression of the known function f (x, y), and assume that the binary function f (x, y) is a continuous differentiable function.

Example Let n = 2, a = 1, f (x, y) = e^{-(x²+y²)/-40}, (x, y) ∈ [0, 1] × [0, 1], and the precision of a given piecewise linear function S approximate to f is σ = 0.1. Please give the realization process of this approximation by the approximation factor.

In fact, by Definition 2.3 and m- mesh subdivision of Note 1, for arbitrary (x, y) ∈ [0, 1] × [0, 1], there are i₁, i₂ $\in ℕ$ such that (x, y) ∈ Δ_{i
₁
i
₂}, and the length of the right angle side of triangle Δ_{i
₁
i
₂} is $\frac{a}{m} = \frac{1}{m}$ , where m is the number of subdivisions waiting to be determined, and each component coordinate on Δ_{i
₁
i
₂} meets $| x_{1} - x_{2} | < \frac{1}{m}$ and $| y_{1} - y_{2} | < \frac{1}{m}$ . See Fig. 1 or Fig. 2 below.

Fig. 2

6-mesh subdivision graph of Δ (1) when n = 2.

Obviously, the function f (x, y) = e^{-(x²+y²)/-40} is uniform continuity on closed set Δ_{i
₁
i
₂}, then, for arbitrary ɛ > 0, there is δ > 0, such that |f (x₁, y₁) - f (x₂, y₂) | < ɛ when for any (x₁, y₁) , (x₂, y₂) ∈ Δ_{i
₁
i
₂} with || (x₁, y₁) - (x₂, y₂) || < δ. Now, if it is satisfies

$\begin{matrix} | | (x_{1}, y_{1}) - (x_{2}, y_{2}) | | & = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}} \\ \leq \sqrt{2 (\frac{1}{m})^{2}} = \frac{\sqrt{2}}{m} < δ, \end{matrix}$ (8) then, we only need to select the natural number with the number of subdivision $m \geq \sqrt{2} / - δ$ .

Next, we will determine the value of the minimum number of subdivision m according to the path σ → ɛ → δ → m in turn with approximation accuracy σ = 0.1.

Firstly, for the above ɛ > 0, n = 2 and given accuracy σ = 0.1, by Theorem 4.1 and Lemma 2, if $\begin{matrix} {∥ S - f ∥}_{\infty} < (\frac{a}{m | | D_{2} | |} \sum_{i = 1}^{2} \sum_{k = 1}^{2} | | B_{k}^{i} | | + 1) ɛ \\ = \frac{n (n + 1) + 2}{2} ɛ = 4 ɛ \leq σ = 0.1 . \end{matrix}$

The solution is ɛ ≤ 0.1/ - 4 = 1/ - 40. So we may take ɛ = 1/ - 40.

Secondly, as f (x, y) is a continuous differentiable function on the closed set Δ_{i
₁
i
₂}, and the partial derivatives of f satisfy $| \frac{\partial f}{\partial x} | = \frac{1}{20} {xe}^{- (x^{2} + y^{2}) / - 40} \leq \frac{1}{20}$ and $| \frac{\partial f}{\partial y} | = \frac{1}{20} {ye}^{- (x^{2} + y^{2}) / - 40} \leq \frac{1}{20}$ .

By the binary Taylor formula, we can expand f (x, y) at point (x₁, y₁) in the first order, that is, for all (x, y) ∈ Δ_{i
₁
i
₂}, there is θ ∈ (0, 1), such that $\begin{matrix} f (x, y) = f (x_{1}, y_{1}) \\ + \frac{\partial f (x_{1} + θ (x - x_{1}), y_{1} + θ (y - y_{1}))}{\partial x} (x - x_{1}) \\ + \frac{\partial f (x_{1} + θ (x - x_{1}), y_{1} + θ (y - y_{1}))}{\partial y} (y - y_{1}) . \end{matrix}$

Let (x, y) = (x₂, y₂), then we have $\begin{matrix} \begin{matrix} | f (x_{2}, y_{2}) - f (x_{1}, y_{1}) | \\ = | \frac{\partial f (x_{1} + θ (x_{2} - x_{1}), y_{1} + θ (y_{2} - y_{1}))}{\partial x} (x_{2} - x_{1}) \\ + \frac{\partial f (x_{1} + θ (x_{2} - x_{1}), y_{1} + θ (y_{2} - y_{1}))}{\partial y} (y_{2} - y_{1}) | \\ \leq | \frac{\partial f (x_{1} + θ (x_{2} - x_{1}), y_{1} + θ (y_{2} - y_{1}))}{\partial x} | | x_{2} - x_{1} | \\ + | \frac{\partial f (x_{1} + θ (x_{2} - x_{1}), y_{1} + θ (y_{2} - y_{1}))}{\partial y} | | y_{2} - y_{1} | . \end{matrix} \end{matrix}$

Especially, when |x₁ - y₁| < δ and |x₂ - y₂| < δ, it is not hard to get that $\begin{matrix} | f (x_{2}, y_{2}) - f (x_{1}, y_{1}) | & \leq \frac{1}{20} (| x_{2} - x_{1} | + | y_{2} - y_{1} |) \\ < \frac{1}{20} (δ + δ) = \frac{δ}{10} . \end{matrix}$

According to the uniform continuity of f (x, y) = e^{-(x²+y²)/-40} on Δ_{i
₁
i
₂}, for ɛ = 1/ - 40 > 0, if $\begin{matrix} | f (x_{2}, y_{2}) - f (x_{1}, y_{1}) | < \frac{δ}{10} \leq \frac{1}{40} = ɛ . \end{matrix}$

The solution is δ ≤ 1/ - 4, we take δ = 1/ - 4. Finally, we can gain the number of subdivision based on formula (8) $m \geq \frac{\sqrt{2}}{δ} = 4 \sqrt{2} \approx 5.656$ . Thus, the minimum number of subdivision m = 6.

Next, we will construct a specific bivariate piecewise linear function on $ℝ^{2}$ with the number of subdivision m = 6 as follows:

Firstly, the compact set [0, 1] × [0, 1] = Δ (1) is divided into 6-meshes, and 36 small squares with side length of 1/ - 6 are obtained. In order to satisfy the rule of three points to determine a plane, then divide each small square into two parts with the diagonal, so as to get 72 small isosceles right angle triangles. We denote they as Δ_{i
₁
i
₂}, where i₁ = 1, 2, ⋯ , 6 ; i₂ = 1, 2, ⋯ , 12. See Fig. 2.

It is not difficult to see that the vertex coordinates of all these small triangles can be determined. For example, $\begin{matrix} Δ_{1, 2} : (0, 0), (\frac{1}{6}, 0), (\frac{1}{6}, \frac{1}{6}); Δ_{2, 3} : (\frac{1}{6}, \frac{1}{6}), (\frac{1}{6}, \frac{1}{3}), \\ (\frac{1}{3}, \frac{1}{3}); Δ_{6, 12} : (\frac{5}{6}, \frac{5}{6}), (1, \frac{5}{6}), (1, 1) . \end{matrix}$

Now, the vertex coordinates of the surface of each triangular prism that each small triangle Δ_j,i on $ℝ^{3}$ under the action of f are also determined in turn. For example, the vertex coordinates of the triangular prism of Δ_2,3 on $ℝ^{3}$ are in turn

$\begin{matrix} (\frac{1}{6}, \frac{1}{6}, e^{- \frac{1}{720}}), (\frac{1}{6}, \frac{1}{3}, e^{- \frac{5}{1440}}), (\frac{1}{3}, \frac{1}{3}, e^{- \frac{1}{180}}) . \end{matrix}$

According to the formulas (5) and (7), it is not difficult to obtain the plane equation determined by these three points. The specific steps are as follows:

$\begin{matrix} \begin{matrix} | D_{2} | = | \begin{matrix} 1 / - 6 & 1 / - 6 & 1 \\ 1 / - 6 & 1 / - 3 & 1 \\ 1 / - 3 & 1 / - 3 & 1 \end{matrix} | = | \begin{matrix} 1 / - 6 & 1 / - 6 & 1 \\ 0 & 1 / - 6 & 0 \\ 1 / - 6 & 1 / - 6 & 0 \end{matrix} | = - \frac{1}{36}, \\ | D_{21}^{3} | = | \begin{matrix} e^{- 1 / - 720} & 1 / - 6 & 1 \\ e^{- 5 / - 1440} & 1 / - 3 & 1 \\ e^{- 1 / - 180} & 1 / - 3 & 1 \end{matrix} | = \frac{1}{6} e^{- \frac{5}{1440}} - \frac{1}{6} e^{- \frac{1}{180}}; \\ | D_{22}^{3} | = | \begin{matrix} 1 / - 6 & e^{- 1 / - 720} & 1 \\ 1 / - 6 & e^{- 5 / - 1440} & 1 \\ 1 / - 3 & e^{- 1 / - 180} & 1 \end{matrix} | = \frac{1}{6} e^{- \frac{1}{720}} - \frac{1}{6} e^{- \frac{5}{1440}}, \\ | D_{23}^{3} | = | \begin{matrix} 1 / - 6 & 1 / - 6 & e^{- 1 / - 720} \\ 1 / - 6 & 1 / - 3 & e^{- 5 / - 1440} \\ 1 / - 3 & 1 / - 3 & e^{- 1 / - 180} \end{matrix} | = \frac{1}{36} e^{- \frac{1}{180}} - \frac{1}{18} e^{- \frac{1}{720}} . \end{matrix}, \end{matrix}$

We will immediately obtain the plane equation determined by Δ_2,3 in $ℝ^{3}$ with plugging these determinant values into the formula (4), it’s not hard to get that $\begin{matrix} S_{2, 3} (x, y) = 6 (e^{- \frac{1}{180}} - e^{- \frac{5}{1440}}) x \\ + 6 (e^{- \frac{5}{1440}} - e^{- \frac{1}{720}}) y + 2 e^{- \frac{1}{720}} - e^{- \frac{1}{180}, (x, y) \in Δ_{2, 3}} . \end{matrix}$

Similarly, the analytic expression of each piecewise linear function of Δ_1,1, Δ_1,2, ⋯ , Δ_2,1, ⋯ , Δ_6,1, ⋯, Δ_6,12 in $ℝ^{3}$ can be determined in turn. Therefore, the analytic expression of the piecewise linear function S (x, y) on [0, 1] × [0, 1] as follows: $S (x, y) = {\begin{matrix} 6 (e^{- \frac{1}{720}} - e^{- \frac{1}{1440}}) x - 6 (1 - e^{- \frac{1}{1440}}) y + 1, (x, y) \in Δ 1, 1, \\ 6 (e^{- \frac{1}{1440}} - 1) x + 6 (e^{- \frac{1}{720}} - e^{- \frac{1}{1440}}) y + 1, (x, y) \in Δ 1, 2, \\ \dots \dots \\ \begin{matrix} 6 (e^{- \frac{1}{180}} - e^{- \frac{5}{1440}}) x + 6 (e^{- \frac{5}{1440}} - e^{- \frac{1}{720}}) y + 2 e^{- \frac{1}{720}} - e^{- \frac{1}{180}}, (x, y) \in Δ 2, 3, \\ 6 (e^{- \frac{61}{1440}} - e^{- \frac{5}{144}}) x + 6 (e^{- \frac{1}{20}} - e^{- \frac{61}{1440}}) y + 6 e^{- \frac{5}{144}} - 5 e^{- \frac{1}{20}}, (x, y) \in Δ 6, 12. \end{matrix} \end{matrix}$

With the analytic expression of the piecewise linear function S (x, y), it is not difficult to draw the spatial surface graph and mixed surface graph of f (x, y) and S (x, y) on [0, 1] × [0, 1] by using MATLAB software programming, as shown in Figs. 3-4.

Fig. 3

Surface graph of a given function f on Δ (1).

Fig. 4

Mixed surface graphs of f and S on Δ (1).

However, only from the mixed Fig. 4, we have not enough reason to say that S can approximate f to arbitrary accuracy. Hence, we will test the approximation ability of the piecewise linear function S by sample points. We may randomly take 30 samples on [0, 1] × [0, 1], and calculate their values and errors at these sample points according to the analytical expressions of f (x, y) and S (x, y), respectively, as follows:

It is not difficult to see from Table 1 that the values of the function f (x, y) and S (x, y) are the same at the vertex coordinates, and their error values D_i are all zero, which is consistent with the condition $S_{i_{1} i_{2} \cdot \cdot \cdot i_{n}} (x_{k}^{*}) = f (x_{k}^{*})$ under the PLF is constructed in Section 2. However, it is not enough to judge that S can approximate f to arbitrary precision according to infinite norm only by randomly selecting the values of these 30 sample points in Table 1.

Table 1

The corresponding values and errors of f (x, y) and S (x, y) at 30 sample points

Number i	Sample point (x₁, x₂)	f (x₁, x₂) = X_i	S (x₁, x₂) = Y_i	D_i = X_i - Y_i
1	(1 / 6, 1/ 6)	0.998612075	0.998612075	0
2	(1/ 3, 1 / 3)	0.994459848	0.994459848	0
3	(1/ 2, 1/ 3)	0.991012850	0.991012850	0
4	(2/ 3, 1/ 6)	0.988263857	0.988263857	0
5	(2/ 3, 1 / 2)	0.982788725	0.982788725	0
6	(5 / 6, 1/ 3)	0.980062544	0.980062544	0
7	(5/ 6, 5 / 6)	0.965873677	0.965873677	0
8	(1/ 6, 1/ 3)	0.996533799	0.996533799	0
9	(1/ 2, 1 / 2)	0.987577800	0.987577800	0
10	(5/6, 2 / 3)	0.971929292	0.971929292	0
11	(1/6, 1/12)	0.999132321	0.998958935	0.000173386
12	(1/4, 1/ 3)	0.995669128	0.995496825	0.000172303
13	(1/12, 1/8)	0.999435923	0.999132486	0.000303437
14	(1/24,1/12)	0.999783010	0.999479468	0.000303542
15	(1/12,1/24)	0.999783010	0.999479468	0.000303542
16	(1/ 8, 1/12)	0.999435923	0.999132487	0.000303436
17	(1/12,1/12)	0.999652838	0.999306037	0.000346801
18	(1/ 6, 1/ 4)	0.997745601	0.996532355	0.001213246
19	(11/12, 5/6)	0.962358674	0.962198631	0.000160043
20	(1, 11 / 12)	0.955035330	0.954876504	0.000476796
21	(1/ 4, 1/ 8)	0.998048781	0.997746368	0.000302413
22	(1/ 4, 1 /24)	0.998395386	0.998092867	0.001302519
23	(11/12,11/12)	0.958856463	0.958551551	0.000304912
24	(1 / 4, 1 / 4)	0.996879878	0.996535960	0.000343918
25	(1/ 12, 5/24)	0.998742111	0.997572213	0.001169898
26	(1/12,23/24)	0.977131852	0.976841083	0.000290769
27	(11/12,23/24)	0.956985524	0.956714027	0.000271497
28	(11/12,21/24)	0.960647666	0.960375091	0.000272575
29	(5/ 6,11 /12)	0.962358674	0.962198631	0.000160043
30	(1/ 4, 1/ 12)	0.998265395	0.997919798	0.000345597

Next, we will apply the t- test method in Statistics to verify that the PLF S can indeed approximate to a continuous function f by the number of subdivision m = 6.

Assuming that the error data D (i) = Y_i - X_i (i = 1, 2, ·· · , 30) in Table 1 are from a sample from the normal population distribution $N (μ_{D}, σ_{D}^{2})$ , where both mean value μ_D and variance $σ_{D}^{2}$ are unknown. According to the t- hypothesis test method in statistical inference, we can easily calculate the values of the mean value $\bar{D}$ and variance s_D as follows: $\begin{matrix} \begin{matrix} \bar{D} = \frac{1}{30} \sum_{i = 1}^{30} D (i) = \frac{0.008520673}{30} = 0.000284022, \\ s_{D} = \sqrt{\frac{1}{30 - 1} \sum_{i = 1}^{30} (D_{i} - \bar{D})^{2}} \\ = \sqrt{\frac{0.0000035988}{29}} \approx 0.000352274 . \end{matrix} \end{matrix}$

For the error data {Y_i - X_i} in Table 1, we can test the hypothesis {H₀, H₁} under the significance level α = 0.05, where the hypothesis (acceptance domain H₀ and rejection domain) satisfies H₀ : μ_D = 0, H₁ : μ_D ≠ 0.

Adopting the approach of the t-test, we select the test statistic $t = (\bar{D} - 0) / - (s_{D} / - \sqrt{n})$ , let n = 30, 1ptμ = 0 and α = 0.05. By looking up to the t-distribution Table for t-hypothesis test, we can obtain t_α (n - 1) = t_0.05 (29) =1.6991. Therefore, the rejection domain of the hypothesis test is $\begin{matrix} t = \frac{\bar{D} 0}{s_{D} / - \sqrt{n}} \geq t_{α} (n 1) = 1.6991 . \end{matrix}$

On the other hand, according to the above mean value $\bar{D}$ and variance s_D we can easily calculate the observational value of t is $\begin{matrix} t = \frac{\bar{D} - 0}{s_{D} / - \sqrt{30}} = \frac{0.000284022}{0.000352274 / - \sqrt{30}} \\ \approx 4.41603003 > 1.6991 . \end{matrix}$

Clearly, the observation value t falls within the rejection region H₁. Hence, we must reject the hypothesis H₀ under significance level α = 0.05. Therefore, the piecewise linear function S can approximate to the continuous function f with arbitrary accuracy.

6 Conclusion

So far, the problem left by Ref. [27] have been successfully solved, and it is verified that PLFs can approach a continuous function. It is not difficult to see from Lemma 2 and Theorem 4.1 that when the PLFs approach an unknown continuous function, the approximation factor is only related to the space dimension n, but not to the the number of subdivision m. Moreover, the larger the n is, the larger the approximation factor is, while the approximation accuracy is reduced. Hence, it is not enough to increase the approximation accuracy of PLFs only by increasing the number of subdivision m. Of course, how to select the vertex coordinates of the polyhedrons is also a key issue when calculating algebraic cofactors of determinant in Theorem 4.1. In three-dimensional space, it may change the approximation factor and affect the approximation accuracy if we select the vertex coordinates according to the left-hand rule. Therefore, it will be the next focus of the study on how to select the optimal vertex coordinates to minimize the approximation factor and improve the approximation accuracy. In addition, whether PLFs can be used to study the approximation of T-S fuzzy systems is also a problem worthy of further discussion.

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