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# 数学代写|数值分析代写Numerical analysis代考|STAT360 Normed linear spaces

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## 数学代写数值分析代写Numerical analysis代考|Normed linear spaces

In order to be able to talk about ‘best approximation’ in a rigorous manner we need to recall from Chapter 2 the concept of norm; this will allow us to compare various approximations quantitatively and select the one which has the smallest approximation error. The definition given in Section $2.7$ applies to a linear space consisting of functions in the same way as to the finite-dimensional linear spaces considered in Chapter 2.
Definition 8.1 Suppose that $\mathcal{V}$ is a linear space over the field $\mathbb{R}$ of real numbers. A nonnegative function $|\cdot|$ defined on $\mathcal{V}$ whose value at $f \in \mathcal{V}$ is denoted by $|f|$ is called a norm on $\mathcal{V}$ if it satisfies the following axioms:
(1) $|f|=0$ if, and only if, $f=0$ in $\mathcal{V}$;
(2) $|\lambda f|=|\lambda||f|$ for all $\lambda \in \mathbb{R}$, and all $f$ in $\mathcal{V}$;
(3) $|f+g| \leq|f|+|g|$ for all $f$ and $g$ in $\mathcal{V}$ (the triangle inequality). A linear space $\mathcal{V}$, equipped with a norm, is called a normed linear space.

## 数学代写|数值分析代写Numerical analysis代考|Best approximation in the ∞-norm

According to the Weierstrass Approximation Theorem any function $f$ in $\mathrm{C}[a, b]$ can be approximated arbitrarily well from the set of all polynomials. Clearly, if instead of the set of all polynomials we restrict ourselves to the set of polynomials $\mathcal{P}n$ of degree $n$ or less, with $n$ fixed, then it is no longer true that, for any $f \in \mathrm{C}[a, b]$ and any $\varepsilon>0$, there exists $p_n \in \mathcal{P}_n$ such that $$\left|f-p_n\right|{\infty}<\varepsilon .$$
Consider, for example, the function $x \mapsto \sin x$ defined on the interval $[0, \pi]$ and fix $n=0$; then $|f-q|_{\infty} \geq 1 / 2$ for any $q \in \mathcal{P}0$, and therefore there is no $q$ in $\mathcal{P}_0$ such that $|f-q|{\infty}<1 / 2$. A similar situation will arise if $\mathcal{P}_0$ is replaced by $\mathcal{P}_n$, with the polynomial degree $n$ fixed. ${ }^1$

It is therefore relevant to enquire just how well a given function $f$ in $\mathrm{C}[a, b]$ may be approximated by polynomials of a fixed degree $n \geq 0$. This question leads us to the following approximation problem.
(A) Given that $f \in \mathrm{C}[a, b]$ and $n \geq 0$, fixed, find $p_n \in \mathcal{P}n$ such that $$\left|f-p_n\right|{\infty}=\inf {q \in \mathcal{P}_n}|f-q|{\infty} ;$$
such a polynomial $p_n$ is called a polynomial of best approximation of degree $n$ to the function $f$ in the $\infty$-norm.

## 数学代写数值分析代写 数学分析代写|规范线性空间

(3) 对于$mathcal{V}$中的所有$f$和$g$，$|f+g| \leq|f|+|g|$（三角不等式）。一个线性空间$mathcal{V}$，配备了一个规范，被称为规范化线性空间。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。