## 数学代写|勒贝格积分代写Lebesgue Integration代考|Math720

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|勒贝格积分代写Lebesgue Integration代考|Continuity and Differentiability

The fourth big question asks for the relationship between continuity and differentiability. We know that a function that is differentiable at a given value of $x$ must also be continuous at that value, and it is clear that the converse does not hold. The function $f(x)=|x|$ is continuous but not differentiable at $x=0$. But how nondifferentiable can a continuous function be?

Throughout the first half of the nineteenth century, it was generally believed that a continuous function would be differentiable at most points. ${ }^4$ Mathematicians recognized that a function might have finitely many values at which it failed to have a derivative. There might even be a sparse infinite set of points at which a continuous function was not differentiable, but the mathematical community was honestly surprised when, in 1875 , Gaston Darboux and Paul du-Bois Reymond ${ }^5$ published examples of continuous functions that are not differentiable at any value.
The question then shifted to what additional assumptions beyond continuity would ensure differentiability. Monotonicity was a natural candidate. Weierstrass constructed a strictly increasing continuous function that is not differentiable at any algebraic number, that is to say, at any number that is the root of a polynomial with rational coefficients. It is not differentiable at $1 / 2$ or $\sqrt{2}$ or $\sqrt[3]{5}-2 \sqrt[21]{35}$. Weierstrass’s function is differentiable at $\pi$. Can we find a continuous, increasing function that is not differentiable at any value? The surprising answer is No. In fact, in a sense that later will be made precise, a continuous, monotonic function is differentiable at “most” values of $x$. There are very important subtleties lurking behind this fourth question.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Term-by-term Integration

Returning to Fourier series, we saw that the heuristic justification relied on interchanging summation and integration, integrating an infinite series of functions by integrating each summand. This works for finite summations. It is not hard to find infinite series for which term-by-term integration leads to a divergent series or, even worse, a series that converges to the wrong value.

Weierstrass had shown that if the series converges uniformly, then term-by-term integration is valid. The problem with this result is that the most interesting series, especially Fourier series, often do not converge uniformly and yet term-by-term integration is valid. Uniform convergence is sufficient, but it is very far from necessary. As we shall see, finding useful conditions under which term-by-term integration is valid is very difficult so long as we cling to the Riemann integral. As Lebesgue would show in the opening years of the twentieth century, his definition of the integral yields a simple, elegant solution, the Lebesgue dominated convergence theorem.

1.1.1. Find the Fourier expansions for $f_1(x)=x$ and $f_2(x)=x^2$ over $[-\pi, \pi]$.
1.1.2. For the functions $f_1$ and $f_2$ defined in Exercise 1.1.1, differentiate each summand in the Fourier series for $f_2$. Do you get the summands in the Fourier series for $2 f_1$ ? Differentiate each summand in the Fourier series for $f_1$. Do you get the summand in the Fourier series for $f_1^{\prime}(x)$ ?
1.1.3. Using the Fourier series expansion for $x^2$ (Exercise 1.1.1) evaluated at $x=\pi$, show that
$$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6} .$$
1.1.4. Show that if $k$ is an integer $\geq 1$, then
$$\int_{-\pi}^\pi \cos (k x) d x=\int_{-\pi}^\pi \sin (k x) d x=0 .$$

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Term-by-term Integration

Weierstrass 已经表明，如果级数一致收敛，则逐项积分是有效的。这个结果的问题在于，最有趣的级 数，尤其是傅立叶级数，通常不会一致收敛，但逐项积分是有效的。均匀收敛就足够了，但远非必要。 正如我们将要看到的，只要我们坚持黎曼积分，就很难找到使逐项积分有效的有用条件。正如勒贝格在 20 世纪初所表明的那样，他对积分的定义产生了一个简单、优雅的解决方案，即勒贝格支配的收敛定 理。
1.1.1. 求傅立叶展开式 $f_1(x)=x$ 和 $f_2(x)=x^2$ 超过 $[-\pi, \pi]$.
1.1.2. 对于函数 $f_1$ 和 $f_2$ 在练习 1.1.1 中定义，区分傅立叶级数中的每个被加数 $f_2$. 你得到傅立叶级数中的 被加数了吗 $2 f_1$ ? 区分傅里叶级数中的每个被加数 $f_1$. 你得到傅立叶级数中的被加数了吗 $f_1^{\prime}(x)$ ?
1.1.3. 使用傅里叶级数展开 $x^2$ (练习 1.1.1) 评估于 $x=\pi$ ， 显示
$$\sum_{n=1}^{\infty} \frac{1}{n^2}=\frac{\pi^2}{6}$$
1.1.4. 证明如果 $k$ 是一个整数 $\geq 1$ ， 然后
$$\int_{-\pi}^\pi \cos (k x) d x=\int_{-\pi}^\pi \sin (k x) d x=0$$

## 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。

## 数学代写|勒贝格积分代写Lebesgue Integration代考|MATH6210

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|勒贝格积分代写Lebesgue Integration代考|The Fundamental Theorem of Calculus

The fundamental theorem of calculus is, in essence, simply a statement of the equivalence of the two means of understanding integration, as the inverse process of differentiation and as a limit of sums of products. The precise theorems to which this designation refers today arise from the assumption that integration is defined as a limiting process. They then clarify the precise relationship between integration and differentiation. The actual statements that we shall use are given by the following theorems.

Theorem 1.1 (FTC, evaluation). If $f$ is the derivative of $F$ at every point on $[a, b]$, then under suitable hypotheses we have that
$$\int_a^b f(t) d t=F(b)-F(a) .$$
Theorem $1.2$ (FTC, antiderivative). If $f$ is integrable on the interval $[a, b]$, then under suitable hypotheses we have that
$$\frac{d}{d x} \int_a^x f(t) d t=f(x) .$$
The first of these theorems tells us how we can use any antiderivative to obtain a simple evaluation of a definite integral. The second shows that the definite integral can be used to create an antiderivative, the definite integral of $f$ from $a$ to $x$ is a function of $x$ whose derivative is $f$. Both of these statements would be meaningless if we had defined the integral as the antiderivative. Their meaning and importance comes from the assumption that $\int_a^b f(t) d t$ is defined as a limit of summations.
In both cases, I have not specified the hypotheses under which these theorems hold. There are two reasons for this. One is that much of the interesting story that is to be told about the creation of analysis in the late nineteenth century revolves around finding necessary and sufficient conditions under which the conclusions hold. When working with Riemann’s definition of the integal, the answer is complicated. The second reason is that the hypotheses that are needed depend on the way we choose to define the integral. For Lebesgue’s definition, the hypotheses are quite different.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|A Brief History of Theorems 1.1 and 1.22

The earliest reference to Theorem $1.1$ of which I am aware is Siméon Denis Poisson’s 1820 Suite du Mémoire sur les Intégrales Définies. There he refers to it as “the fundamental proposition of the theory of definite integrals.” Poisson’s work is worth some digression because it illustrates the importance of how we define the definite integral and the difficulties encountered when it is defined as the difference of the values of an antiderivative at the endpoints.

Siméon Denis Poisson (1781-1840) studied and then taught at the École Polytechnique. He succeeded to Fourier’s professorship in mathematics when Fourier departed for Grenoble to become prefect of the department of Isère. It was Poisson who wrote up the rejection of Fourier’s Theory of the Propoagation of Heat in Solid Bodies in 1808. When, in 1815, Poisson published his own article on the flow of heat, Fourier pointed out its many flaws and the extent to which Poisson had rediscovered Fourier’s own work.

Poisson, as a colleague of Cauchy at the École Polytechnique, almost certainly was aware of Cauchy’s definition of the definite integral even though Cauchy had not yet published it. But the relationship between Poisson and Cauchy was far from amicable, and it would have been surprising had Poisson chosen to embrace his colleague’s approach. Poisson defines the definite integral as the difference of the values of the antiderivative. It would seem there is nothing to prove. What Poisson does prove is that if $F$ has a Taylor series expansion and $F^{\prime}=f$, then
$$F(b)-F(a)=\lim {n \rightarrow \infty} \sum{j=1}^n t f(a+(j-1) t), \quad \text { where } t=\frac{b-a}{n} .$$
Poisson begins with the observation that for $1 \leq j \leq n$ and $t=(b-a) / n$, there is a $k \geq 1$ and a collection of functions $R_j$ such that
$$F(a+j t)=F(a+(j-1) t)+t f(a+(j-1) t)+t^{1+k} R_j(t),$$
and therefore
\begin{aligned} F(b)-F(a) & =\sum_{j=1}^n[F(a+j t)-F(a+(j-1) t)] \ & =\sum_{j=1}^n t f(a+(j-1) t)+t^{1+k} \sum_{j=1}^n R_j(t) . \end{aligned}

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Fundamental Theorem of Calculus

$$\int_a^b f(t) d t=F(b)-F(a)$$

$$\frac{d}{d x} \int_a^x f(t) d t=f(x)$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|A Brief History of Theorems 1.1 and 1.22

Siméon Denis Poisson (1781-1840) 在巴黎综合理工学院学习并任教。当傅立叶前往格勒诺布尔成为 伊泽尔省省长时，他继承了傅立叶的数学教授职位。正是泊松在 1808 年写下了拒绝傅立叶的固体热传播 理论的文章。在 1815 年，泊松发表了他自己关于热流的文章时，傅立叶指出了它的许多缺陷以及泊松在 多大程度上重新发现了傅里叶自己的工作。

$$F(b)-F(a)=\lim n \rightarrow \infty \sum j=1^n t f(a+(j-1) t), \quad \text { where } t=\frac{b-a}{n} .$$

$$F(a+j t)=F(a+(j-1) t)+t f(a+(j-1) t)+t^{1+k} R_j(t),$$

$$F(b)-F(a)=\sum_{j=1}^n[F(a+j t)-F(a+(j-1) t)] \quad=\sum_{j=1}^n t f(a+(j-1) t)+t^{1+k} \sum_{j=1}^n R_j(t)$$

## 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。

## 数学代写|勒贝格积分代写Lebesgue Integration代考|MAT00013H

statistics-lab™ 为您的留学生涯保驾护航 在代写勒贝格积分Lebesgue Integration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。我们的专家在代写勒贝格积分Lebesgue Integration代写方面经验极为丰富，各种代写勒贝格积分Lebesgue Integration相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

Fourier’s method for expanding an arbitrary function $F$ defined on $[-\pi, \pi]$ into a trigonometric series is to use integration to calculate coefficients:
\begin{aligned} & a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \ & b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) . \end{aligned}
The Fourier expansion is then given by
$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$
The heuristic argument for the validity of this procedure is that if $F$ really can be expanded in a series of the form given in Equation (1.3), then
\begin{aligned} \int_{-\pi}^\pi & F(x) \cos (n x) d x \ = & \int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x \ = & \int_{-\pi}^\pi \frac{a_0}{2} \cos (n x) d x+\sum_{k=1}^{\infty} \int_{-\pi}^\pi a_k \cos (k x) \cos (n x) d x \ & +\sum_{k=1}^{\infty} \int_{-\pi}^\pi b_k \sin (k x) \cos (n x) d x \end{aligned}
Since $n$ and $k$ are integers, all of the integrals are zero except for the one involving $a_n$. These integrals are easily evaluated:
$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

Fourier and Cauchy were among the first to fully realize the inadequacy of defining integration as the inverse process of differentiation. It is too restrictive. Fourier wanted to apply his methods to arbitrary functions. Not all functions have antiderivatives that can be expressed in terms of standard functions. Fourier tried defining the definite integral of a nonnegative function as the area between the graph of the function and the $x$-axis, but that begs the question of what we mean by area. Cauchy embraced Leibniz’s understanding as a limit of products, and he found a way to avoid infinitesimals.

To define $\int_a^b f(x) d x$, Cauchy worked with finite approximating sums. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we consider
$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$
If we can force all of these approximating sums to be as close to each as other as we wish simply by limiting the size of the difference between consecutive values in the partition, then these summations have a limiting value that is designated as the value of the definite integral, and the function $f$ is said to be integrable over $[a, b]$.

Equipped with this definition, Cauchy succeeded in proving that any continuous or piecewise continuous function is integrable. The class of functions to which Fourier’s analysis could be applied was suddenly greatly expanded.

When Riemann turned to the study of trigonometric series, he wanted to know the limits of Cauchy’s approach to integration. Was there an easy test that could be used to determine whether or not a function could be integrated? Cauchy had chosen to evaluate the function at the left-hand endpoint of the interval simply for convenience. As Riemann thought about how far this definition could be pushed, he realized that his analysis would be simpler if the definition were stated in a slightly more complicated but essentially equivalent manner. Given a partition of $[a, b]$ : $\left(a=x_0<x_1<\cdots<x_n=b\right)$, we assign a tag to each interval, a number $x_j^$ contained in that interval, and consider all sums of the form $$\sum_{k=1}^n f\left(x_k^\right)\left(x_k-x_{k-1}\right) \text {. }$$

# 勒贝格积分代考

## 数学代写|勒贝格积分代写Lebesgue Integration代考|The Five Big Questions

$$a_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \cos (k x) d x \quad(k \geq 0), \quad b_k=\frac{1}{\pi} \int_{-\pi}^\pi F(x) \sin (k x) d x \quad(k \geq 1) .$$

$$F(x)=\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right] .$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\int_{-\pi}^\pi\left(\frac{a_0}{2}+\sum_{k=1}^{\infty}\left[a_k \cos (k x)+b_k \sin (k x)\right]\right) \cos (n x) d x=\int_{-\pi}^\pi \frac{a_0}{2}$$

$$\int_{-\pi}^\pi F(x) \cos (n x) d x=\pi a_n .$$

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Cauchy and Riemann Integrals

$$\sum_{k=1}^n f\left(x_{k-1}\right)\left(x_k-x_{k-1}\right) .$$

## 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。