## 数学代写|傅里叶分析代写Fourier analysis代考|AMTH246

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
• (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|Continuous, Discrete, and Digital Signals

This type of classification characterizes the type of sampling of the dependent and independent variables. Sampling the amplitude is called quantization. Table $1.1$ shows the signal classification based on sampling the amplitude and time. When both the variables of a signal can assume continuum of values, it is called a continuous signal, such as the ambient temperature. Most of the naturally occurring signals are of this type. The temperature measured by a digital thermometer is a quantized continuous signal. This type of signals occurs in the reconstruction of a continuous signal from its sampled version. Sampled continuous-valued signal is a discrete signal. This type of signals, shown in Fig. 1.4c, d, is used in the analysis of discrete signals and systems. A quantized discrete signal is called a digital signal, used in the digital systems.

The sinusoidal signals are defined by the values of the coordinates on a circle in Fig. 1.3. In each rotation of a point on the circle, the same set of values are produced indefinitely. This type of signals, such as the sine and cosine functions, is periodic signals. While only one period of a periodic signal contains new information, periodicity is required to represent signals such as power and communication signals. In communication engineering, the message signal is aperiodic and the carrier signal is periodic. Finite duration signals are represented, by the practically most often used version of the Fourier analysis, assuming periodic extension. The finite signal is considered as the values of one period and concatenation of it indefinitely on either side yields a periodic signal. A signal $x(t)$ is said to be periodic, if $x(t)=x(t+T)$, for all values of $t$ from $-\infty$ to $\infty$ and $T>0$ is a positive constant. The minimum value of $T$ that satisfies the constraint is the period. A periodic signal shifted by an integral number of its period remains unchanged. A signal that is not periodic is aperiodic, such as the impulse, step and ramp signals shown in Fig. 1.1 and the real exponential. The period is infinity, so that there is no indefinite repetition. The everlasting definition of a periodic signal is for mathematical convenience. In practice, physical devices are switched on at some time and the response reaches a steady state, after the transient response dies down.

## 数学代写|傅里叶分析代写Fourier analysis代考|Even- and Odd-Symmetric Signals

Any signal can be decomposed into its even and odd components. Knowing whether a signal is even or odd may reduce computational and storage requirements in its processing. If a signal $x(t)$ satisfies the condition
$$x(-t)=x(t) \text { for all } t$$ then it is said to be even. The plot of such a signal is symmetrical about the vertical axis at the origin. For example, the cosine waveforms, shown in Figs. 1.4a and 1.6b, are even. For the signal in Fig. 1.6b,
$$0.5 \cos \left(\frac{2 \pi}{32}(-n)\right)=0.5 \cos \left(\frac{2 \pi}{32} n\right)$$
If a signal $x(t)$ satisfies the condition
$$x(-t)=-x(t) \text { for all } t,$$
then it is said to be odd. The plot of such a signal is antisymmetrical about the vertical axis at the origin. For example, the sine waveforms, shown in Figs. 1.4b and 1.6c, are odd. For the signal in Fig. 1.6c,
$$\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32}(-n)\right)=-\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32} n\right)$$
Any function can be decomposed into its even and components. Let the even and odd components of $x(t)$ be $x_e(t)$ and $x_o(t)$, respectively. Then,
$$x(t)=x_e(t)+x_o(t) \text { and } x(-t)=x_e(t)-x_o(t)$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Even- and Odd-Symmetric Signals

$$x(-t)=x(t) \text { for all } t$$

$$0.5 \cos \left(\frac{2 \pi}{32}(-n)\right)=0.5 \cos \left(\frac{2 \pi}{32} n\right)$$

$$x(-t)=-x(t) \text { for all } t,$$

$$\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32}(-n)\right)=-\frac{\sqrt{3}}{2} \sin \left(\frac{2 \pi}{32} n\right)$$

$$x(t)=x_e(t)+x_o(t) \text { and } x(-t)=x_e(t)-x_o(t)$$

## 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|傅里叶分析代写Fourier analysis代考|MAT3105

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
• (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|Sinusoids and Complex Exponentials

The impulse and the sinusoid are the two most important signals in signal and system analysis. The impulse is the basis for convolution and the sinusoid is the basis for transfer function. The cosine and sine functions are two of the most important functions in trigonometry. As these functions are the basis functions in Fourier analysis, we have study them in detail.

The unit circle, defined by $x^2+y^2=1$ and shown in Fig. 1.3, is a circle with its center located at the origin and radius 1 . For each point on the circle defined by the coordinates $(x, y)$, starting at $(1,0)$ and moving in the counterclockwise direction, with $\theta \geq 0$ (the angle subtended by the $x$-axis and the line joining the point and the origin), the sine (sin) and cosine (cos) functions are defined in terms of its coordinates $(x, y)$ as
$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$
If the point lies on a circle of radius $r$, then
$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$
Clearly, the sinusoids are of periodic nature. Any function defined on a circle will be a periodic function of an angular variable $\theta$. Therefore, the trigonometric functions are also called circular functions. The argument $\theta$ is measured in radians or degrees. The radian is defined as the angle subtended between the $x$-axis and the line between the point and the origin on the unit circle. One radian is defined as the angle subtended by unit arc length. Since the circumference of the unit circle is $2 \pi$, one complete revolution is $2 \pi \mathrm{rad}$. In degree measure, $2 \pi=360^{\circ}$ and $\pi=180^{\circ}$. One radian is approximately $180 / \pi=57.3^{\circ}$.

A linear combination of sine and cosine functions is a sinusoid, in rectangular form, given by
$$a \cos (\theta)+b \sin (\theta)$$
where $a$ and $b$ are real numbers with $a \neq 0$ or $b \neq 0$. With $c=\sqrt{a^2+b^2}$, and $\cos (d)=a / c$ and $\sin (d)=b / c$,
$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$
is called the polar form of the sinusoid.

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

By using sine and cosine functions, signals can be represented. But it involves two basic functions and the two associated constants. It is found that an equivalent representation of signals is obtained using the complex exponential function, in which only one basic function and one associated constant is involved. The compact representation and the ease of manipulating the exponential functions make its use mandatory in the analysis of signals and systems. However, practical devices generate sine and cosine functions. Euler’s formula is the bridge between the theory and the practice. With $b$ any positive real number except 1 ,
$$x(t)=b^t$$
is called the exponential function with base $b$. Our primary interest, in this book, is the complex exponential function of the form
$$x(\theta)=A e^{j \theta}$$
The base is $e$, which is approximately $2.71828$. The exponent is a complex number with its real part zero (pure imaginary number). The coefficient of the exponential $A$ is a complex number.

The exponential $e^{j \theta}$, shown in Fig. 1.5, is a unit rotating vector, rotating in the counterclockwise direction. The exponential carries the same information about a sinusoid in an equivalent form, which is advantageous in the analysis of signals and systems. In combination with the exponential $e^{-j \theta}$, which rotates in the clockwise direction, a real sinusoidal waveform can be obtained. Since
$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$
solving for $\cos (\theta)$ and $\sin (\theta)$ results in
$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Sinusoids and Complex Exponentials

$$\cos (\theta)=x \quad \text { and } \quad \sin (\theta)=y$$

$$\cos (\theta)=x / r \text { and } \sin (\theta)=y / r, \quad r=\sqrt{x^2+y^2}$$

$$a \cos (\theta)+b \sin (\theta)$$

$$a \cos (\theta)+b \sin (\theta)=c \cos (\theta-d)$$

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

$$x(t)=b^t$$

$$x(\theta)=A e^{j \theta}$$

$$e^{j \theta}=\cos (\theta)+j \sin (\theta) \text { and } e^{-j \theta}=\cos (\theta)-j \sin (\theta),$$

$$\cos (\theta)=\frac{e^{j \theta}+e^{-j \theta}}{2} \text { and } \sin (\theta)=\frac{e^{j \theta}-e^{-j \theta}}{j 2}$$

## 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|傅里叶分析代写Fourier analysis代考|MAST20026

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
• (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|Unit-Impulse Signal

The unit-impulse and the sinusoidal signals are the most important signals in the study of signals and systems. The continuous unit-impulse $\delta(t)$ is a signal with a shape and amplitude such that its integral at the point $t=0$ is unity. It is defined, in terms of an integral, as
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$
It is assumed that $x(t)$ is continuous at $t=0$ so that the value $x(0)$ is distinct. The product of $x(t)$ and $\delta(t)$ is
$$x(t) \delta(t)=x(0) \delta(t)$$
since the impulse exists only at $t=0$. Therefore,
$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$
The value of the function $x(t)$, at $t=0$, is sifted out or sampled by the defining operation. By using shifted impulses, any value of $x(t)$ can be sifted.

It is obvious that the integral of the unit-impulse is the unit-step. Therefore, the derivative of the unit-step signal is the unit-impulse signal. The value of the unit-step is zero for $t<0$ and 1 for $t>0$. Therefore, the unit area of the unit-impulse, as the derivative of the unit-step, must occur at $t=0$. The unit-impulse and the unitstep signals enable us to represent and analyze signals with discontinuities as we do with continuous signals. For example, these signals model the commonly occurring situations such as opening and closing of switches.

The continuous unit-impulse $\delta(t)$ is difficult to visualize and impossible to realize in practice. However, the approximation of it by some functions is effective in practice and can be used to visualize its effect on signals and its properties. While there are other functions that approach an impulse in the limit, the rectangular function is often used to approximate the impulse. The unit-impulse, for all practical purposes, is essentially a narrow rectangular pulse with unit area. Suppose we compress it by a factor of 2 , the area, called its strength, becomes $1 / 2=0.5$. The scaling property of the impulse is given as
$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$
With $a=-1, \delta(-t)=\delta(t)$ implying that the impulse is an even-symmetric signal. For example,
$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$
The discrete unit-impulse signal, shown in Fig. 1.1a, is defined as
$$\delta(n)=\left{\begin{array}{l} 1 \text { for } n=0 \ 0 \text { for } n \neq 0 \end{array}\right.$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal

The discrete unit-step signal, shown in Fig. 1.1b, is defined as
$$u(n)=\left{\begin{array}{l} 1 \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$ For positive values of its argument, the value of the unit-step signal is unity and it is zero otherwise. An arbitrary function can be expressed in terms of appropriately scaled and shifted unit-step or impulse signals. By this way, any signal can be specified, for easier mathematical analysis, by a single expression, valid for all $n$. For example, a pulse signal, shown in Fig. 1.2a, with its only nonzero values defined as $\{x(1)=1, x(2)=1, x(3)=1\}$ can be expressed as the sum of the two delayed unitstep signals shown in Fig. 1.2b, $x(n)=u(n-1)-u(n-4)$. The pulse can also be represented as a sum of delayed impulses. $$x(n)=u(n-1)-u(n-4)=\sum_{k=1}^3 \delta(n-k)=\delta(n-1)+\delta(n-2)+\delta(n-3)$$ The continuous unit-step signal is defined as $$u(t)= \begin{cases}1 & \text { for } t>0 \ 0 & \text { for } t<0 \\ \text { undefined for } t=0\end{cases}$$ The value $u(0)$ is undefined and can be assigned a suitable value from 0 to 1 to suit a specific problem. In Fourier analysis, $u(0)=0.5$. A common application of the unit-step signal is that multiplying a signal with it yields the causal form of the signal. For example, the continuous signal $\sin (t)$ is defined for $-\infty0$.

The discrete unit-ramp signal, shown in Fig. 1.1c, is also often used in the analysis of signals and systems. It is defined as
$$r(n)=\left{\begin{array}{l} n \text { for } n \geq 0 \ 0 \text { for } n<0 \end{array}\right.$$
It linearly increases for positive values of its argument and is zero otherwise.
The three signals, the unit-impulse, the unit-step, and the unit-ramp, are related by the operations of sum and difference. The unit-impulse signal $\delta(n)$ is equal to $u(n)-u(n-1)$, the first difference of the unit-step. The unit-step signal $u(n)$ is equal to $\sum_{k=0}^{\infty} \delta(n-k)$, the running sum of the unit-impulse. The shifted unit-step signal $u(n-1)$ is equal to $r(n)-r(n-1)$. The unit-ramp signal $r(n)$ is equal to
$$r(n)=n u(n)=\sum_{k=0}^{\infty} k \delta(n-k) .$$
Similar relations hold for continuous type of signals.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Impulse Signal

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0)$$

$$x(t) \delta(t)=x(0) \delta(t)$$

$$\int_{-\infty}^{\infty} x(t) \delta(t) d t=x(0) \int_{-\infty}^{\infty} \delta(t) d t=x(0)$$

$$\delta(a t)=\frac{1}{|a|} \delta(t), a \neq 0$$

$$\delta(3 t-1)=\delta\left(3\left(t-\frac{1}{3}\right)\right)=\frac{1}{3} \delta\left(t-\frac{1}{3}\right)$$

$\$ \$$Idelta(n)=Veft { 1 for n=00 for n \neq 0 正确的。 ## 数学代写|傅里叶分析代写Fourier analysis代考|Unit-Step Signal 如图 1.1b 所示，离散单位阶跃信号定义为 \ \$$
$\mathrm{u}(\mathrm{n})=\backslash \mathrm{left}{$
1 for $n \geq 00$ for $n<0$ 、正确的。 Forpositivevaluesofitsargument, thevalueoftheunit – stepsignalisunityanditiszer $x(n)=u(n-1)-u(n-4)=\backslash$ sum_ ${k=1}^{\wedge} 3$ \delta(nk)=ldelta(n-1)+ldelta(n-2)+ \三角洲 $(n-3)$ Thecontinuousunit – stepsignalisdefinedas $u(t)=$ $$\left{\begin{array}{l} 1 \ \text { undefined for } t=0 \end{array} \text { for } t>00 \quad \text { for } t<0\right.$$
$\$ \$$价值 u(0) 是末定义的，可以分配一个从 0 到 1 的合适值以适应特定问题。在傅立叶分析中， u(0)=0.5. 单位阶跃信号的一个常见应用是将一个信号与其相乘产生信号的因果形式。例如，连续信 号 \sin (t) 被定义为 -\infty 0. 离散单位斜坡信号，如图 1.1c 所示，也经常用于信号和系统的分析。它被定义为 \ \$$
$$r(n)=\backslash l \text { eft }{$$
$n$ for $n \geq 00$ for $n<0$

## 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|傅里叶分析代写Fourier analysis代考|AMTH246

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
• (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|A First Sketch of the Argument

We start by recalling, very briefly, the usual approach taken in proving Roth’s Theorem. One takes a set $A$ of density $\delta$ in $\mathbb{Z} / N \mathbb{Z}$, and compares the number of length 3 Arithmetic Progressions in $A$ with $\frac{1}{2} \delta^3 N^2$. This is roughly the number of 3-term APs in a random subset of $\mathbb{Z} / N \mathbb{Z}$. The difference $D$ between these two quantities can be expressed using the Fourier Coefficients $\hat{A}(r)$ of $A$. If $D$ is small then $A$ contains a progression of length 3 becuase it approximates a random set. Otherwise $D$ is large, and we can deduce that some $\hat{A}(r)$ is large for $r \neq 0$. This information in turn allows us to deduce that $A$ has increased density $\delta+c \delta^2$ in some reasonably large Arithmetic Progression $P$. But $P$ is affinely equivalent to ${1, \ldots, N}$, and so we can iterate the argument. However one can only increment the density $O\left(\delta^{-1}\right)$ times before it becomes greater than 1, which is clearly impossible. Hence if $A$ is large enough then it contains a 3 -term AP.
Bourgain’s point of departure seems to be the following. Suppose that
$$\hat{A}(r)=\sum_n A(n) e^{2 \pi i n r / N}$$
is large. To show that $A$ has increased density in some progression $P$, one has to somehow get rid of the exponential terms appearing here. In the usual proof of Roth’s Theorem this is done by splitting up $\mathbb{Z} / N \mathbb{Z}$ into small progressions on which $e^{2 \pi i n r / N}$ is roughly constant as $n$ varies. This, however, is rather inefficient – rather a lot of small progressions are required. Suppose instead that one forgets about progressions, and splits $\mathbb{Z} / N \mathbb{Z}$ up into sets on which $|n r / N|$ is roughly constant. We could easily deduce that $A$ has increased density on one of these sets. Unfortunately however this information is not equivalent to the original hypothesis, since one of the new sets is not affinely equivalent to ${1, \ldots, N}$. Hence we have to strengthen the entire hypothesis that we are trying to prove.

The “sets” that we are discussing here are of course just translates of Bohr Neighbourhoods. Hence we shall try to prove something like the following.

Conjecture 4 Let $A$ be a subset of some Bohr Neighbourhood $\Lambda$, such that $|A|=\delta|\Lambda|$. Then for fixed $\delta$ and “sufficiently large” $\Lambda, A$ contains a three-term Arithmetic Progression.

Since $\mathbb{Z} / N \mathbb{Z}$ is trivially a Bohr Neighbourhood, we might hope that this would imply Roth’s Theorem with a better bound.

There are many difficulties to overcome in order to make the above idea work, as we shall discover. These stem principally from three facts.

## 数学代写|傅里叶分析代写Fourier analysis代考|Definitions and Elementary Properties

We begin by defining what we mean by a Bohr Neighbourhood from now on.
Definition 5 Let $\theta=\left{\theta_1, \ldots, \theta_d\right} \in \mathbb{R}^d$, and let $\epsilon$ and $M$ be real numbers with $\epsilon<\frac{1}{2}$. Then we define the Bohr Neighbourhood $\Lambda_{\theta, \epsilon, M}$ to be the set of all $n \in \mathbb{Z}$ such that $|n| \leq M$ and $\left|n \theta_j\right| \leq \epsilon$ for $j=1, \ldots, d$.

This is clearly very similar to the “mod $N$ ” version of the same name. We take the opportunity to record here some simple facts about Bohr Neighbourhoods which will be useful later.
Lemma $6\left|\Lambda_{\theta, \epsilon, M}\right| \geq \epsilon^d M$
Proof Let $\mathbb{S}^d$ be the unit torus $\mathbb{R}^d / \mathbb{Z}^d$. Consider the set of all $P_n=\left(\left|n \theta_1\right|, \ldots,\left|n \theta_d\right|\right) \in \mathbb{S}^d$ for integers $n \in[1, M]$. This has size $M$, so some $\epsilon$-cube $\mathcal{B}$ of $\mathbb{S}^d$ contains at least $M \epsilon^d$ of the $P_i$ (this “obvious” averaging argument actually requires careful analysis its justification). Let $\mathcal{C}$ be the set of all $n \in[1, M]$ for which $P_n \in \mathcal{B}$. Then there is an injection
$$\phi: \mathcal{C} \rightarrow \Lambda_{\theta, \epsilon, M}$$
defined by $\phi(n)=n-n_0$, where $n_0 \in \mathcal{C}$ is arbitrary.
Lemma $7\left|\Lambda_{\theta, \epsilon, M}\right|<8^{d+1}\left|\Lambda_{\theta, \frac{\epsilon}{2}, \frac{M}{2}}\right|$
Proof Divide $\Lambda_{\theta, \epsilon, M}$ into sets $A_i$ such that
(i) $\left{\left(\left|n \theta_1\right|, \ldots,\left|n \theta_d\right|\right) \mid n \in A_i\right}$ is contained in an $\frac{\epsilon}{2}$-cube in $\mathbb{S}^d$;
(ii) $A_i$ is contained in an interval of length $\frac{M}{2}$.
This can be achieved with $8^{d+1}$ sets $A_i$. Each $A_i$ injects to $\Lambda_{\theta, \frac{5}{2}, \frac{M}{2}}$ by sending $n$ to $n-n_0$, where $n_0 \in A_i$ is arbitrary. The result follows.

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|A First Sketch of the Argument

Bourgain 的出发点似乎是以下几点。假设
$$\hat{A}(r)=\sum_n A(n) e^{2 \pi i n r / N}$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Definitions and Elementary Properties

$$\phi: \mathcal{C} \rightarrow \Lambda_{\theta, \epsilon, M}$$

(二) $A_i$ 包含在长度区间内 $\frac{M}{2}$.

## 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|傅里叶分析代写Fourier analysis代考|MAT3105

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
• (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|Random translations

In particular, if $\mu(E) \sim 1 / N$, then we can find $N$ translates $g_1 E, \ldots, g_N E$ of $E$ whose union has measure $\sim 1$, thus these translates behave as if they are disjoint “up to constants”. We observe that the same claim also holds for any homogeneous space $G / H$ of a compact group $G$ (with the attendant Haar probability measure), simply by lifting subsets of that homogeneous space back up to $G$.

Lemma $5.1$ allows one in many cases to reduce the analysis of “small” subsets of a compact group $G$ (or a homogenous space $G / H$ of $G$ ) to the analysis of “large” sets, particularly if the problem in question enjoys some sort of translation symmetry with respect to the group $G$. This idea was for instance famously exploited by Stein [28] in his maximum principle equating almost everwhere convergence results for translation-invariant operators with weak-type $(p, p)$ maximal inequalities. In $[6, \S 6]$, Bourgain noted that these techniques could also be combined with the factorization theory of Pisier, Nikishin, and Maurey [24] (which Bourgain had previously used, for instance, in [13]), although it has subsequently been realized that the arguments can be formulated without explicit reference to that theory. Specifically, in the context of restriction estimates for the sphere, Bourgain observed

Proposition 5.2. Suppose that $d \geq 2$ and $1<p<2$ is such that one has the restriction estimate
$$|\hat{f}|_{L^1\left(S^{d-1}, d \sigma\right)} \lesssim p p, a|f|_{L^p\left(\mathbb{K}^d\right)}$$
for all Schwartz functions $f: \mathbb{R}^d \rightarrow \mathbb{C}$, where $\sigma$ is normalized surface measure on the sphere $S^d 1$. Then ane ran automatirally improve this to the stmnger patimate
$$|\hat{f}|_{L^{p, \infty}\left(S^{d-1}, d \sigma\right)} \lesssim p, d|f|_{L^p\left(\mathbb{R}^d\right)} .$$

Proof (Sketch). We can normalize $|f|_{L^P\left(\mathbb{R}^d\right)}=1$. Let $\lambda>0$, and let $E \subset S^{d-1}$ denote the level set
$$E:=\left{\omega \in S^{d-1}:|\hat{f}(\omega)| \geq \lambda\right}$$

## 数学代写|傅里叶分析代写Fourier analysis代考|Local Structure of Sets

A very common type of argument in Bourgain’s paper is the following. One has finite sets $A, B \subseteq \mathbb{Z}$, and a function $f: \mathbb{Z} \rightarrow \mathbb{R}$ for which, say,
$$\sum_{n \in A} f(n) \geq \eta|A| .$$
One then wishes to conclude that there is some translate $B^{\prime}=B+m$ for which
$$\sum_{n \in B^{\prime}} f(n) \geq(1-\epsilon) \eta|B|$$
for some small $\epsilon$. This section is devoted to an exploration of situations under which such a principle holds. One feels that the principle is doomed to failure unless $B$ is much “smaller” than $A$ (unless $B$ equals $A$, of course). However one also needs $A$ to “look like $B$ locally” to avoid examples such as $A={0,5,10, \ldots, 5(n-1)}, B={0,1,2,3,4}$. In such an example the behaviour of $f$ on $A$ gives very little information on the behaviour of $f$ on translates of $B$.
After some thought the following definition seems natural.
Definition 1 Let
$$Q(n)=|{m \in A \mid n \in B+m}|=|(n-B) \cap A| .$$
Then we say that $A$ looks $\kappa$-locally like $B$ if
$$\sum_n|Q(n)-A(n)| B | \leq \kappa|A||B| .$$
To relate this to Bourgain’s paper, we note that if $\alpha$ and $\beta$ are the characteristic measures associated to the sets $A$ and $B$ then $A$ looks $\kappa$-locally like $B$ precisely when
$$|\alpha * \beta-\alpha|_1 \leq \kappa .$$
Let us now see how this definition relates to the type of averaging argument discussed above. Let $f: \mathbb{Z} \rightarrow \mathbb{R}$ be a function with $|f|_{\infty} \leq 1$, and suppose that $A$ looks $\kappa$-locally like $B$. Then
\begin{aligned} \left|\sum_{m \in A} \sum_{n \in B+m} f(n)-\right| B\left|\sum_{n \in A} f(n)\right| & =\left|\sum_n(Q(n)-|B| A(n)) f(n)\right| \ & \leq \kappa|A||B| . \end{aligned}
Hence we have the following Lemma.

# 傅里叶分析代写

## 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。