Posted on Categories:Lebesgue Integration, 勒贝格积分, 数学代写

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## 数学代写|勒贝格积分代写Lebesgue Integration代考|Uniform and Pointwise Convergence

Throughout the text we will be interested in the following question: If a sequence of functions $\left{f_n\right}$ “converges” to a limit function $f$ does the sequence of numbers $\left{\int_a^b f_n(x) d x\right}$ converge to a limit equal to the integral of the limit function? Put another way, we are interested in when lim and $\int$ commute, i.e., when is
$$\lim {n \rightarrow \infty} \int_a^b f_n(x) d x=\int_a^b \lim {n \rightarrow \infty} f_n(x) d x ?$$
The answer, as we will see, depends on what we mean by the sequence of functions “converging,” i.e., what does $\lim f_n$ mean. It turns out there are many interesting (and very different) choices for what we might mean. Among the types of convergence that we will consider the strongest is called uniform convergence. We recall its definition.
Definition 1.4.1. (Uniform convergence). A sequence of functions $\left{f_m\right}$ is said to converge uniformly on $[a, b]$ to a function $f$ if for every $\varepsilon>0$ there is an $M$ (independent of $x$ ) such that for all $x \in[a, b]$
$$\left|f(x)-f_m(x)\right|<\varepsilon \text { whenever } m \geq M .$$
We contrast this with the following much weaker notion of convergence.

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

We now want to define the integral of a more general class of functions than just step functions. Since we know how to integrate step functions it is natural to try to take a sequence of better and better step function approximations to a more general function $f$ and define the integral of $f$ to be the limit of the integrals of the approximating step functions. For this to work we need to know that the limit of the integrals exists and that it does not depend on the choice of approximating step functions. It turns out that all of this works if the more general function $f$ can be uniformly approximated by step functions, i.e., if there is a sequence of step functions which converges uniformly to $f$. As is typical in mathematics when we have a collection of objects which behave in a way we like we make it into a definition.

Definition 1.5.1. (Regulated function). A function $f:[a, b] \rightarrow \mathbb{R}$ is called regulated provided there is a sequence $\left{f_m\right}$ of step functions which converges uniformly to $f$.

Another way to state this is to say a regulated function is one which can be uniformly approximated as closely as we wish by a step function. We can now prove that the limit of the integrals of the approximating step functions always exists and does not depend on the choice of approximating step functions.

## 数学代写|勒贝格积分代写Lebesgue Integration代考|Uniform and Pointwise Convergence

Missing or unrecognized delimiter for \left “收敛 “到一个等于极限函数积分的极限？换句话说，我们对$lim$和$int$的换算感兴趣，也就是说，何时是
$$\lim n \rightarrow \infty \int_a^b f_n(x) d x=\int_a^b \lim n \rightarrow \infty f_n(x) d x ?$$

$$\left|f(x)-f_m(x)\right|<\varepsilon\text { whenever } m \geq M .$$

## 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。

Posted on Categories:数学代写, 表示论

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## 数学代写|表示论代写Representation Theory代考|The Key Point Is the Convergence

We keep all previous notation. In this last section we shall see that the key point is the convergence of the integral
$$\left(I_{\mathfrak{h}2 \mathfrak{h}_1}^G \varphi\right)(g):=\left(I{\mathfrak{h}2 \mathfrak{h}_1} \varphi\right)(g)=\oint{H_2 /\left(H_1 \cap H_2\right)} \varphi(g h) \chi_f(h) \Delta_{H_2, G}^{-1 / 2}(h) d v(h)(g \in G)$$
At least formally, it is clear that the function $I_{\mathfrak{h}2 \mathfrak{h}_1} \varphi$ satisfies the necessary covariance condition for belonging in the space $\mathscr{H}{\pi_2}$, and also that $I_{\mathfrak{h}2 \mathfrak{h}_1}$ commutes with left translations. We can therefore assert that the convergence of the integral is one major issue to deal with. We call elements of $I(f, \mathfrak{g})$ Pukanszky polarizations. In this section we look at all pairs $\left(\mathfrak{h}_1, \mathfrak{h}_2\right)$ of Pukanszky polarizations at $f$ which meet the following convergence property: $(C P)$ : The integral $$\oint{H_2 /\left(H_1 \cap H_2\right)}|\varphi(g h)| \Delta_{H_2, G}^{-\frac{1}{2}}(h) d v(h), g \in G$$
converges on a $G$-invariant and $\mathscr{U}(\mathfrak{g})$-invariant dense subspace $\mathscr{H}$ of $\mathscr{H}_{\pi_1}^{\infty}$.

## 数学代写|表示论代写Representation Theory代考|Notation and Backgrounds

Let $G$ be a connected and simply connected nilpotent Lie group with Lie algebra $\mathfrak{g}$. Let exp denote, as earlier, the exponential map, so that $G=\exp \mathfrak{g}$. Let $V, W$ be real vector spaces of finite dimension with $W \subset V$. We denote by $V^{\star}$ the dual vector space of $V$ and by $W^{\perp}$ the orthogonal to $W$ in $V^{\star}$. If $u_1, \ldots, u_p,(p \in \mathbb{N})$ indicate linearly independent vectors in $V$, we denote by $\mathbb{R}$-span $\left(u_1, \ldots, u_p\right)$ the vector subspace of $V$ they span, and we say the basis $\left{u_1, \ldots, u_p\right}$ generates this space. Given $l \in \mathfrak{g}^{\star}$ and $X \in \mathfrak{g}$, we denote by $\langle l, X\rangle$ the image of $X$ under $l$. Recall that the kernel of the bilinear form $B_l$, defined in Sect. 1.2.4 by $B_l(X, Y)=$ $\langle l,[X, Y]\rangle$, is denoted by $\mathfrak{g}(l)=\left{X \in \mathfrak{g} ; B_l(X, Y)=0\right.$ for all $\left.Y \in \mathfrak{g}\right}$. The largest ideal contained in $\mathfrak{g}(l)$ will be denoted by $\mathfrak{a}(l)$. Clearly
$$\mathfrak{a}(l)=\bigcap_{\phi \in G \cdot l} \mathfrak{g}(\phi) .$$
Let $\mathfrak{h} \in S(l, \mathfrak{g})$ and let $\chi_l$ be the unitary character of the analytic subgroup $H=$ $\exp h$ associated to $l$ by
$$\chi_l(\exp X)=e^{-i\langle l, X\rangle}$$
for $X \in \mathfrak{h}$.

## 数学代写|表示论代写Representation Theory代考|关键点在于收敛性

$$\left(I_{mathrm{h} 2\mathrm{~h}1}^G\varphi\right)(g)。 =left(I\mathfrak{h} 2\mathfrak{h}_1 \varphi\right)(g)=oint H_2 /\left(H_1 \cap H_2\right) \varphi(g h) \chi_f(h) δ{H_2, G}^{-1 / 2}(h) d v(h) (g\in G)$$

$B_l(X, Y)=\langle l,[X, Y]\rangle$，用Missing或未识别的分隔符表示〈left〉。$mathfrak{g}(l)$中包含的最大理想将用$mathfrak{a}(l)$来表示。显然
$$mathfrak{a}(l)=\bigcap_{phi\subset G-l}。\mathfrak{g}(phi) 。$$

\chi_l(exp X)=e^{-i(l, X\rangle}。

for $X\in mathfrak{h}$ 。

## 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。