Posted on Categories:Mathematical Analysis, 数学代写, 数学分析

# 数学代写|数学分析作业代写Mathematical Analysis代考|MATH7400 Fourier Series and Orthogonal Polynomials

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Fourier Series and Orthogonal Polynomials

In section $3.7$ we studied the geometry of inner product spaces more than their metric properties. We now have a bigger toolbox with which we can tackle inner product spaces. Before we pose the central questions of this section, let us summarize the highlights of section 3.7, upon which this section rests heavily. Let $\left{u_{1}, u_{2}, \ldots\right}$ be an infinite orthonormal sequence of vectors in an inner product space $H$. The orthogonal projection of an element $x \in H$ on the finite-dimensional space $M_{n}=\operatorname{Span}\left(\left{u_{1}, \ldots, u_{n}\right}\right)$ is, by definition, the vector $S_{n} x=\sum_{i=1}^{n}\left\langle x, u_{i}\right\rangle u_{i}$. We know from theorem 3.7.6 that the vector $S_{n} x$ is the closest vector in $M_{n}$ to $x$, and we also say that $S_{n} x$ is the best approximation of $x$ in $M_{n}$. Now that we have studied convergence in metric spaces, it is natural to ask whether $\lim {n} S{n} x=x$. Unfortunately, we are still not in a position to state an exact set of conditions under which a general answer can be provided because the answer depends on the space $H$ and the sequence $\left{u_{1}, u_{2}, \ldots\right}$. The reader should suspect that completeness is relevant here, and it is. The spaces we study in this section are not complete, and this is precisely the reason we cannot decisively settle the question posed above about the convergence of the sequence $S_{n} x$. In two of the major examples we consider in this section, we will answer this question satisfactorily but not completely. The full picture will materialize in sections $7.2$ and $8.9$.
Fourier series
In section 3.7, we defined the inner product $\langle f, g\rangle=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(x) \overline{g(x)} d x$ on the space $\mathcal{C}[-\pi, \pi]$. The sequence
$$\left{u_{n}(t)=e^{i n t}: n \in \mathbb{Z}\right}$$
is an orthonormal sequence with respect to the above inner product. The norm of a function $f$ induced by the inner product will be denoted by $|f|_{2}$ in order to distinguish it from the uniform norm on $\mathcal{C}[-\pi, \pi]$, which will also play a prominent role in this section. Thus the uniform norm of a function $f \in \mathcal{C}[a, b]$ will be denoted by the usual notation $|f|_{\infty}$, while
$$|f|_{2}=\left(\frac{1}{2 \pi} \int_{-\pi}^{\pi}|f(x)|^{2} d x\right)^{1 / 2} .$$
It is clear that $|f|_{2} \leq|f|_{\infty}$.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|The Tchebychev Polynomials

In this special case, we take
$$(a, b)=(-1,1)$$
and
$$\omega(x)=\frac{1}{\sqrt{1-x^{2}}} .$$
Observe that the space $H$ of square integrable functions with respect to $\omega$ contains the entire space $\mathcal{C}[-1,1]$

A simple and direct derivation of the orthogonal polynomials is possible because of the observation that, for an integer $n \geq 0, \cos (n x)$ can be expressed as a polynomial of $\cos x$. For example, $\cos (2 x)=2 \cos ^{2} x-1$. The next lemma proves the existence of such polynomials and establishes the three-term recurrence relation among them.

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|Fourier Series and Orthogonal Polynomials

〈left 的分隔符缺失或无法识别 是内积空间中向量的无限正交序列 $H$. 元膆的正交投影 $x \in H$ 在有限维空间上 \left 的分隔符胡失或无法识别 是，根居定义，向量 $S_{n} x=\sum_{i=1}^{n}\left\langle x, u_{i}\right\rangle u_{i}$. 我们从定理 $3.7 .6$ 知道向量
$S_{n} x$ 是最近的向量 $M_{n}$ 至 $x$ ，我们也说 $S_{n} x$ 是的最佳近似值 $x$ 在 $M_{n}$. 现在我们已经研究了度量空间中的收敛性，很自然地要问是否 $\lim n S n x=x$. 不幸的是，我们仍然无法说明可以提供一般箜䅁的确切条件，因为答案取决于空间 $H$ 和序列
\left 的分隔符缺失或无法识别 . 读者应该怀疑完整性在这里是相关的，而且确实如此。本节我们研究的空间 并不完整，这也正是我们无法果断解决上述关于序列收敛的问题的原因 $S_{n} x$. 在本节中我们考虑的两个主要示例中，我们将满意但

《left 的分隔符缺失或无法识别

$$|f|{2}=\left(\frac{1}{2 \pi} \int{-\pi}^{\pi}|f(x)|^{2} d x\right)^{1 / 2} .$$

## 数学代写|数学分析作业代写MATHEMATICAL ANALYSIS代考|The Tchebychev Polynomials

$$(a, b)=(-1,1)$$

$$\omega(x)=\frac{1}{\sqrt{1-x^{2}}} .$$

## 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。