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数学代写|数学分析作业代写Mathematical Analysis代考|MATH7400 Approximation

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数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Approximation

In this section, we prove a large collection of approximation theorems. The highlights include approximating $\mathfrak{Q}^p$ functions by simple functions and continuous functions of compact support. We prove that trigonometric polynomials are dense in $\mathfrak{Q}^2(-\pi, \pi)$, which is the last piece of information we need to settle the question of the convergence of Fourier series of functions in $\mathfrak{Q}^2(-\pi, \pi)$. The important operation of convoluting functions makes its first debut in this section. Finally, we study approximations by $C^{\infty}$ functions, prove the $C^{\infty}$ version of Urysohn’s lemma, and prove that $\mathcal{C}_c^{\infty}\left(\mathbb{R}^n\right)$ is dense in $\mathfrak{Q} p\left(\mathbb{R}^n\right)$.

Lemma 8.7.1 (the Tietze extension theorem). Let $K$ be a compact subset of $\mathbb{R}^n$ and let $f: K \rightarrow[0,1]$ be continuous. Then $f$ can be extended to a continuous function $g \in \mathcal{C}_c\left(\mathbb{R}^n\right)$ such that $0 \leq g \leq 1$. If $K$ is contained in an open set $U$, then $g$ can be constructed in such a way that $\operatorname{supp}(g) \subseteq U$.

Proof. Let $W$ be an open set such that $\bar{W}$ is compact and $K \subseteq W \subseteq \bar{W} \subseteq U$. Let $K_1=$ $f^{-1}([0,1 / 3])$, and $K_2=f^{-1}([2 / 3,1])$. Applying lemma $8.4 .1$ to the closed sets $E=$ $K_1 \cup\left(\mathbb{R}^n-W\right)$, and $F=K_2$, produces a continuous function $g_1: \mathbb{R}^n \rightarrow[0,1 / 3]$ such that $g_1(E)=0$, and $g_1(F)=1 / 3$. By construction, $\operatorname{supp}\left(g_1\right) \subseteq \bar{W}$, and $0 \leq$ $f-g_1 \leq 2 / 3$ on $K$. Applying the same construction to the function $f-g_1$, we can find a function $g_2: \mathbb{R}^n \rightarrow\left[0, \frac{1}{3} \cdot \frac{2}{3}\right]$ such that $\operatorname{supp}\left(g_2\right) \subseteq \bar{W}$, and $0 \leq f-g_1-g_2 \leq$ $\left(\frac{2}{3}\right)^2$ on the set $K$. Continuing this construction yields a sequence of continuous functions $g_i$ on $\mathbb{R}^n$ such that $\operatorname{supp}\left(g_i\right) \subseteq \bar{W}, 0 \leq g_i \leq \frac{2^{i-1}}{3^i}$, and $0 \leq f-g_1-g_2 \cdots-$ $g_i \leq\left(\frac{2}{3}\right)^i$ on $K$. The sequence $G_i=g_1+\ldots+g_i$ is supported in $\bar{W}$ and is Cauchy in the uniform norm on the compact set $\bar{W}$. Therefore $G_i$ converges uniformly to a continuous function $g$. Since $0 \leq f-G_i \leq\left(\frac{2}{3}\right)^i$ on $K, g=f$ on $K$. Because each $G_i$ is supported in $\bar{W} \subseteq U$, so is $g$.

数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Approximation by $\mathcal{C}^{\infty}$ Functions

Definition. For Lebesgue measurable functions $f$ and $g$ on $\mathbb{R}^n$, the convolution of $f$ and $g$ is the function
$$(f * g)(x)=\int_{\mathbb{R}^n} f(x-y) g(y) d y .$$
It is clear that if $(f * g)(x)$ is finite, then $(f * g)(x)=(g * f)(x)$. Thus
$$(f * g)(x)=\int_{\mathbb{R}^n} f(x-y) g(y) d y=\int_{\mathbb{R}^n} f(y) g(x-y) d y .$$
A variety of conditions can be imposed on $f$ and $g$ to guarantee the finiteness of the integral, at least for a.e. $x \in \mathbb{R}^n$. We take for granted the measurability of the function $f(x-y) g(y)$.

In this subsection, we will limit the functions $f$ and $g$ to be continuous functions of compact support. The reader can look at the section exercises for a slightly expanded discussion of the properties of convolutions.

数学代写|数学分析代写MATHEMATICAL ANALYSIS代考|Approximation by $\mathcal{C}^{\infty}$ 功能

$$(f * g)(x)=\int_{\mathbb{R}^n} f(x-y) g(y) d y .$$

$$(f * g)(x)=\int_{\mathbb{R}^n} f(x-y) g(y) d y=\int_{\mathbb{R}^n} f(y) g(x-y) d y .$$

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。