# 数学代写|实分析作业代写Real analysis代考|Uniform Convergence and Differentiation

#### Doug I. Jones

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## 数学代写|实分析作业代写Real analysis代考|Uniform Convergence and Differentiation

In this section, we consider the question of interchange of limits and differentiation. Example 8.1.2(e) shows that even if the sequence $\left{f_n\right}$ converges uniformly to $f$, this is not sufficient for convergence of the sequence $\left{f_n^{\prime}\right}$ of derivatives. Example 8.5.3 will further demonstrate very dramatically the failure of the interchange of limits and differentiation. There we will give an example of a series, each of whose terms has derivatives of all orders, that converges uniformly to a continuous function $f$, but for which $f^{\prime}$ fails to exist at every point of $\mathbb{R}$. Clearly, uniform convergence of the sequence $\left{f_n\right}$ is not sufficient. What is required is uniform convergence of the sequence $\left{f_n^{\prime}\right}$.
THEOREM 8.5.1 Suppose $\left{f_n\right}$ is a sequence of differentiable functions on $[a, b]$. If
(a) $\left{f_n^{\prime}\right}$ converges uniformly on $[a, b]$, and
(b) $\left{f_n\left(x_o\right)\right}$ converges for some $x_o \in[a, b]$,
then $\left{f_n\right}$ converges uniformly to a function $f$ on $[a, b]$, with
$$f^{\prime}(x)=\lim _{n \rightarrow \infty} f_n^{\prime}(x)$$
Remarks. (a) Convergence of $\left{f_n\left(x_o\right)\right}$ at some $x_o \in[a, b]$ is required. For example, if we let $g_n(x)=f_n(x)+n$, then $g_n^{\prime}(x)=f_n^{\prime}(x)$, but $\left{g_n(x)\right}$ need not converge for any $x \in[a, b]$. In Exercise 1 you will be asked to show that uniform convergence of $\left{f_n^{\prime}\right}$ is also required; pointwise convergence is not sufficient.
(b) If in addition to the hypotheses we assume that $f_n^{\prime}$ is continuous on $[a, b]$, then a much shorter and easier proof can be provided using the fundamental theorem of calculus. Since $f_n^{\prime}$ is continuous, by Theorem 6.3.2
$$f_n(x)=f_n\left(x_o\right)+\int_{x_o}^x f_n^{\prime}(t) d t$$
for all $x \in[a, b]$. The result can now be proved using Corollary 8.3.2 and Theorem 8.4.1. The details are left to the exercises (Exercise 2).

## 数学代写|实分析作业代写Real analysis代考|The Weierstrass Approximation Theorem

In this section, we will prove the following well known theorem of Weierstrass.
THEOREM 8.6.1 (Weierstrass) If $f$ is a continuous real-valued function on $[a, b]$, then given $\epsilon>0$, there exists a polynomial $P$ such that
$$|f(x)-P(x)|<\epsilon$$
for all $x \in[a, b]$
An equivalent version, and what we will actually prove, is the following:
If $f$ is a continuous real-valued function on $[a, b]$, then there exists a sequence $\left{P_n\right}$ of polynomials such that
$$f(x)=\lim _{n \rightarrow \infty} P_n(x) \quad \text { uniformly on }[a, b] .$$
Before we prove Theorem 8.6.1, we state and prove a more fundamental result that will also have applications later. Prior to doing so, we need the following definitions.

DEFINITION 8.6.2 A real-valued function $f$ on $\mathbb{R}$ is periodic with period $p$ if
$$f(x+p)=f(x) \quad \text { for all } x \in \mathbb{R} .$$

The canonical examples of periodic functions are the functions $\sin x$ and $\cos x$, both of which are periodic of period $2 \pi$. The graph of a periodic function of period $p$ is illustrated in Figure 8.3. The graphs of a periodic function of period $p$ on any two successive intervals of length $p$ are identical. It is clear that if $f$ is periodic of period $p$, then
$$f(x+k p)=f(x) \quad \text { for all } k \in \mathbb{Z} \text {. }$$

# 实分析代写

## 数学代写|实分析作业代写Real analysis代考|Uniform Convergence and Differentiation

8.5.3 将进一步非常戏剧性地证明极限和微分互换的失 败。在那里我们将给出一个序列的例子，它的每一项都 有所有阶数的导数，它一致地收敛到一个连续函数 $f$ ，但 为此 $f^{\prime}$ 不存在于每一点 $\mathbb{R}$. 显然，序列的一致收敛

$$f^{\prime}(x)=\lim {n \rightarrow \infty} f_n^{\prime}(x)$$ $x_o \in[a, b]$ 是必须的。例如，如果我们让 $g_n(x)=f_n(x)+n$ ，然后 $g_n^{\prime}(x)=f_n^{\prime}(x)$ ，但 左 $\left{\mathrm{g} _\mathrm{n}(\mathrm{x}) \backslash\right.$ 右 $}$ 不需要收敛于任何 $x \in[a, b]$. 在练习 1 是必需的；逐点收敛是不够的。 (b) 如果除了假设之外我们假设 $f_n^{\prime}$ 是连续的 $[a, b]$ ，然后 可以使用微积分基本定理提供更短和更容易的证明。自 从 $f_n^{\prime}$ 是连续的，由定理 6.3.2 $$f_n(x)=f_n\left(x_o\right)+\int{x_o}^x f_n^{\prime}(t) d t$$

## 数学代写|实分析作业代写Real analysis代考|The Weierstrass Approximation Theorem

$$|f(x)-P(x)|<\epsilon$$

$$f(x)=\lim _{n \rightarrow \infty} P_n(x) \quad \text { uniformly on }[a, b] \text {. }$$

$$f(x+p)=f(x) \quad \text { for all } x \in \mathbb{R} .$$

$$f(x+k p)=f(x) \quad \text { for all } k \in \mathbb{Z} \text {. }$$

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## 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。

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