### 数学代写|傅里叶分析代写Fourier analysis代考|MATH3205

statistics-lab™ 为您的留学生涯保驾护航 在代写傅里叶分析Fourier analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写傅里叶分析Fourier analysis代写方面经验极为丰富，各种代写傅里叶分析Fourier analysis相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|傅里叶分析代写Fourier analysis代考|Pointwise Convergence

In the following we will see that for frequently appearing classes of functions stronger convergence results can be proved. A function $f: \mathbb{T} \rightarrow \mathbb{C}$ is called piecewise continuously differentiable, if there exist finitely many points $0 \leq x_{0}<$ $x_{1}<\ldots<x_{n-1}<2 \pi$ such that $f$ is continuously differentiable on each subinterval $\left(x_{j}, x_{j+1}\right), j=0, \ldots, n-1$ with $x_{n}=x_{0}+2 \pi$, and the left and right limits $f\left(x_{j} \pm 0\right), f^{\prime}\left(x_{j} \pm 0\right)$ for $j=0, \ldots, n$ exist and are finite. In the case $f\left(x_{j}-0\right) \neq f\left(x_{j}+0\right)$, the piecewise continuously differentiable function $f: \mathbb{T} \rightarrow \mathbb{C}$ has a jump discontinuity at $x_{j}$ with jump height $\left|f\left(x_{j}+0\right)-f\left(x_{j}-0\right)\right|$. Simple examples of piecewise continuously differentiable functions $f: \mathbb{T} \rightarrow \mathbb{C}$ are the sawtooth function and the rectangular pulse function (see Examples $1.9$ and 1.10). This definition is illustrated in Fig. 1.9.

The next convergence statements will use the following result of RiemannLebesgue.

Lemma $1.27$ (Lemma of Riemann-Lebesgue) Let $f \in L_{1}(\overline{(a, b)})$ with $-\infty \leq$ $a<b \leq \infty$ be given. Then the following relations hold:
$$\lim {|v| \rightarrow \infty} \int{a}^{b} f(x) \mathrm{e}^{-\mathrm{i} x v} \mathrm{~d} x=0,$$
$$\lim {|v| \rightarrow \infty} \int{a}^{b} f(x) \sin (x v) \mathrm{d} x=0, \quad \lim {|v| \rightarrow \infty} \int{a}^{b} f(x) \cos (x v) \mathrm{d} x=0 .$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Uniform Convergence

A useful criterion for uniform convergence of the Fourier series of a function $f \in$ $C(\mathbb{T})$ is the following:
Theorem 1.37 If $f \in C(\mathbb{T})$ fulfills the condition
$$\sum_{k \in \mathbb{Z}}\left|c_{k}(f)\right|<\infty,$$
then the Fourier series of $f$ converges uniformly to $f$. Each function $f \in C^{1}(\mathbb{T})$ has the property (1.49).
Proof By the assumption (1.49) and
$$\left|c_{k}(f) \mathrm{e}^{\mathrm{i} k \cdot}\right|=\left|c_{k}(f)\right|,$$
the uniform convergence of the Fourier series follows from the Weierstrass criterion of uniform convergence. If $g \in C(\mathbb{T})$ is the sum of the Fourier series of $f$, then we obtain for all $k \in \mathbb{Z}$
$$c_{k}(g)=\left\langle g, \mathrm{e}^{\mathrm{i} k \cdot}\right\rangle=\sum_{n \in \mathbb{Z}} c_{n}(f)\left\langle\mathrm{e}^{\mathrm{i} n \cdot}, \mathrm{e}^{\mathrm{i} k \cdot}\right\rangle=c_{k}(f)$$
such that $g=f$ by Theorem $1.1$.
Assume that $f \in C^{1}(\mathbb{T})$. By the convergence Theorem $1.34$ of Dirichlet-Jordan we know already that the Fourier series of $f$ converges uniformly to $f$. This could be also seen as follows: By the differentiation property of the Fourier coefficients in Lemma 1.6, we have $c_{k}(f)=(\mathrm{i} k)^{-1} c_{k}\left(f^{\prime}\right)$ for all $k \neq 0$ and $c_{0}\left(f^{\prime}\right)=0$. By Parseval equality of $f^{\prime} \in L_{2}(\mathbb{T})$ it follows
$$\left|f^{\prime}\right|^{2}=\sum_{k \in \mathbb{Z}}\left|c_{k}\left(f^{\prime}\right)\right|^{2}<\infty .$$
Using Cauchy-Schwarz inequality, we get finally
\begin{aligned} \sum_{k \in \mathbb{Z}}\left|c_{k}(f)\right| &=\left|c_{0}(f)\right|+\sum_{k \neq 0} \frac{1}{|k|}\left|c_{k}\left(f^{\prime}\right)\right| \ & \leq\left|c_{0}(f)\right|+\left(\sum_{k \neq 0} \frac{1}{k^{2}}\right)^{1 / 2}\left(\sum_{k \neq 0}\left|c_{k}\left(f^{\prime}\right)\right|^{2}\right)^{1 / 2}<\infty . \end{aligned}
This completes the proof.

## 数学代写|傅里叶分析代写Fourier analysis代考|Pointwise Convergence

$$\lim |v| \rightarrow \infty \int a^{b} f(x) \mathrm{e}^{-\mathrm{i} x v} \mathrm{~d} x=0$$
$$\lim |v| \rightarrow \infty \int a^{b} f(x) \sin (x v) \mathrm{d} x=0, \quad \lim |v| \rightarrow \infty \int a^{b} f(x) \cos (x v) \mathrm{d} x=0 .$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Uniform Convergence

$$\sum_{k \in \mathbb{Z}}\left|c_{k}(f)\right|<\infty$$

$$\left|c_{k}(f) \mathrm{e}^{\mathrm{i} k \cdot}\right|=\left|c_{k}(f)\right|$$

$$c_{k}(g)=\left\langle g, \mathrm{e}^{\mathrm{i} k \cdot}\right\rangle=\sum_{n \in \mathbb{Z}} c_{n}(f)\left\langle\mathrm{e}^{\mathrm{i} n \cdot}, \mathrm{e}^{\mathrm{i} k \cdot}\right\rangle=c_{k}(f)$$

$$\left|f^{\prime}\right|^{2}=\sum_{k \in \mathbb{Z}}\left|c_{k}\left(f^{\prime}\right)\right|^{2}<\infty .$$

$$\sum_{k \in \mathbb{Z}}\left|c_{k}(f)\right|=\left|c_{0}(f)\right|+\sum_{k \neq 0} \frac{1}{|k|}\left|c_{k}\left(f^{\prime}\right)\right| \quad \leq\left|c_{0}(f)\right|+\left(\sum_{k \neq 0} \frac{1}{k^{2}}\right)^{1 / 2}\left(\sum_{k \neq 0}\left|c_{k}\left(f^{\prime}\right)\right|^{2}\right)^{1 / 2}<\infty$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。