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• 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|信息论作业代写information theory代考|A “Lucky Accident”

The solutions $\left{\varphi_{n}(x)\right}$ of the integral equation (2.57) can be obtained by solving the differential equation
$$\frac{d}{d x}\left(1-x^{2}\right) \frac{d \varphi(x)}{d x}+\left(\chi-c_{0}^{2} x^{2}\right) \varphi(x)=0, \quad|x|<1 .$$
This follows by defining the differential and integral operators
\begin{aligned} &\mathcal{P} \varphi=\frac{d}{d x}\left(1-x^{2}\right) \frac{d \varphi}{d x}-c_{0}^{2} x^{2} \varphi, \ &\mathcal{Q} \varphi=\int_{-1}^{1} \frac{\sin c_{0}(x-y)}{\pi(x-y)} \varphi(y) d y, \end{aligned}
and noticing the commutative property
$$\mathcal{Q P} \varphi=\mathcal{P} \mathcal{Q} \varphi .$$
Since commuting operators admitting a complete set of eigenfunctions share the same eigenfunctions, it follows that the solutions of (2.60) are also solutions of (2.57). Slepian refers to this as a “lucky accident” that allowed him and his collaborators to find the solution to the concentration problem in terms of that of a well-known equation in physics.

The solutions of (2.60) arise in the context of the wave equation, and are known as the prolate spheroidal wave functions of the first kind and of order zero. They are a set of solutions of the Helmholtz equation when this is expressed in a suitable coordinate system. The corresponding eigenvalues are positive, discrete reals $\chi_{0} \leq \chi_{1} \leq \chi_{2} \leq \cdots$. When the eigenfunctions are indexed by increasing values of $\chi$, they agree with the notation of indexing by decreasing values of $\lambda$ used above. It follows that the prolate spheroidal wave functions are the most concentrated, orthogonal, bandlimited functions, and enjoy many properties useful for applications in mathematical physics. Among those, one key property is that their energy falls off sharply beyond a critical phase transition point corresponding to values of the indexes in the neighborhood of $N_{0}=\Omega T / \pi$. This leads to the notion of the asymptotic dimensionality of the space of bandlimited signals.

## 数学代写|信息论作业代写information theory代考|Most Concentrated Functions

The eigenfunctions $\left{\varphi_{n}(x)\right}$ of (2.57) are also well defined for all $x \in \mathbb{R}$. It can be shown that they are orthonormal in $L^{2}(-\infty, \infty)$ and complete in $\mathscr{B}{1}$ there, as well as orthogonal and complete in $L^{2}(-1,1)$, as already noted. They have exactly $n$ zeros in $(-1,1)$, and they are even or odd as $n$ is even or odd. We have \begin{aligned} &\int{-\infty}^{\infty} \varphi_{n}(x) \varphi_{m}(x) d x= \begin{cases}1 & \text { if } n=m \ 0 & \text { otherwise }\end{cases} \ &\int_{-1}^{1} \varphi_{n}(x) \varphi_{m}(x) d x= \begin{cases}\lambda_{n} & \text { if } n=m \ 0 & \text { otherwise. }\end{cases} \end{aligned}
The notation conceals the fact that both the $\left{\varphi_{n}\right}$ and the $\left{\lambda_{n}\right}$ depend on the parameter $c_{0}=\Omega T / 2$. When necessary, we write $\lambda_{n}=\lambda_{n}\left(c_{0}\right), \varphi_{n}(x)=\varphi_{n}\left(c_{0}, x\right)$. It turns out that $\left{\lambda_{n}\left(c_{0}\right)\right}$ are continuous functions of $c_{0}$, and for fixed $c_{0}$ they fall off rapidly with increasing $n$, once $n$ exceeds the Nyquist number $N_{0}=2 c_{0} / \pi=\Omega T / \pi$. This is the phase transition behavior referred to above, which allows the determination of the exact dimension of the space of bandlimited signals spanned by the eigenfunctions $\left{\varphi_{n}\right}$. Another important property is that the Fourier transform of $\varphi_{n}(x)$ restricted to $|x|<1$ has the same form as $\varphi_{n}(x)$, except for a scale factor (see also Section $2.6 .2$ below), namely
$$j^{n} \sqrt{\frac{\lambda_{n}}{2 \pi c_{0}}} \varphi_{n}(x)=\frac{1}{2 \pi} \int_{-1}^{1} \varphi_{n}(s) \exp \left(j c_{0} x s\right) d s$$

# 信息论代写

## 数学代写|信息论作业代写information theory代考|A “Lucky Accident”

$$\frac{d}{d x}\left(1-x^{2}\right) \frac{d \varphi(x)}{d x}+\left(\chi-c_{0}^{2} x^{2}\right) \varphi(x)=0, \quad|x|<1 .$$

$$\mathcal{P} \varphi=\frac{d}{d x}\left(1-x^{2}\right) \frac{d \varphi}{d x}-c_{0}^{2} x^{2} \varphi, \quad \mathcal{Q} \varphi=\int_{-1}^{1} \frac{\sin c_{0}(x-y)}{\pi(x-y)} \varphi(y) d y$$

$$\mathcal{Q} \mathcal{P} \varphi=\mathcal{P} \mathcal{Q} \varphi$$

(2.60) 的解出现在波动方程的上下文中，并且被称为第一类和零阶的长球面波函数。当在合适的坐标系中表示时，它们是亥姆霍兹方程的一组解。相应的特征值 是正的离散实数 $\chi_{0} \leq \chi_{1} \leq \chi_{2} \leq \cdots$ 当特征函数通过增加的值来索引时 $\chi$ ，他们同意通过减少值的索引表示法 $\lambda$ 上面用过。由此可见，长球面波函数是最集中 的、正交的、带限的函数，并且具有许多对数学物理应用有用的性质。其中，一个关键特性是它们的能量在临界相变点之后急剧下降，该钿界相变点对应于附近 的指数值。 $N_{0}=\Omega T / \pi$. 这导致了带限信号空间的渐近维数的概念。

## 数学代写|信息论作业代写information theory代考|Most Concentrated Functions

$\int-\infty^{\infty} \varphi_{n}(x) \varphi_{m}(x) d x=\left{1 \quad\right.$ if $n=m 0 \quad$ otherwise $\quad \int_{-1}^{1} \varphi_{n}(x) \varphi_{m}(x) d x=\left{\lambda_{n} \quad\right.$ if $n=m 0 \quad$ otherwise.

lleft 的分隔符缺失或无法识别

$$j^{n} \sqrt{\frac{\lambda_{n}}{2 \pi c_{0}}} \varphi_{n}(x)=\frac{1}{2 \pi} \int_{-1}^{1} \varphi_{n}(s) \exp \left(j c_{0} x s\right) d s$$

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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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