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数学代写|实分析代写Real Analysis代考|Weierstrass Approximation Theorem

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数学代写|实分析代写Real Analysis代考|Weierstrass Approximation Theorem

We saw as an application of Proposition 1.49 that the function $|x|$ on $[-1,1]$ is the uniform limit of an explicit sequence $\left{P_n\right}$ of polynomials with $P_n(0)=0$. This is a special case of a theorem of Weierstrass that any continuous complex-valued function on a bounded interval is the uniform limit of polynomials on the interval.
The device for proving the Weierstrass theorem for a general continuous complex-valued function is to construct the approximating polynomials as the result of a smoothing process, known as the use of an “approximate identity.” The idea of an approximate identity is an important one in analysis and will occur several times in this book. If $f$ is the given function, the smoothing is achieved by “convolution”
$$\int f(x-t) \varphi(t) d t$$
of $f$ with some function $\varphi$, the integrals being taken over some particular intervals. The resulting function of $x$ from the convolution turns out to be as “smooth” as the smoother of $f$ and $\varphi$. In the case of the Weierstrass theorem, the function $\varphi$ will be a polynomial, and we shall arrange parameters so that the convolution will automatically be a polynomial.
To see how a polynomial $\int f(x-t) \varphi(t) d t$ might approximate $f$, one can think of $\varphi$ as some kind of mass distribution; the mass is all nonnegative if $\varphi \geq 0$. The integration produces a function of $x$ that is the “average” of translates $x \mapsto f(x-t)$ of $f$, the average being computed according to the mass distribution $\varphi$. If $\varphi$ has total mass 1 , i.e., total integral 1 , and most of the mass is concentrated near $t=0$, then $f$ is being replaced essentially by an average of its translates, most of the translates being rather close to $f$, and we can expect the result to be close to $f$.
For the Weierstrass theorem, we use a single starting $\varphi_1$ at stage 1 , namely $c_1\left(1-x^2\right)$ on $[-1,1]$ with $c_1$ chosen so that the total integral is 1 . The graph of $\varphi_1$ is a familiar inverted parabola, with the appearance of a bump centered at the origin. The function at stage $n$ is $c_n\left(1-x^2\right)^n$, with $c_n$ chosen so that the total integral is 1. Graphs for $n=3$ and $n=30$ appear in Figure 1.1. The bump near the origin appears to be more pronounced at $n$ increases, and what we need to do is to translate the above motivation into a proof.

数学代写|实分析代写Real Analysis代考|Fourier Series

A trigonometric series is a series of the form $\sum_{n=-\infty}^{\infty} c_n e^{i n x}$ with complex coefficients. The individual terms of the series thus form a doubly infinite sequence, but the sequence of partial sums is always understood to be the sequence $\left{s_N\right}_{N=0}^{\infty}$ with $s_N(x)=\sum_{n=-N}^N c_n e^{i n x}$. Such a series may also be written as
$$\frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n \cos n x+b_n \sin n x\right)$$
by putting
\begin{aligned} & \left.\begin{array}{rl} e^{i n x} & =\cos n x+i \sin n x \ e^{-i n x} & =\cos n x-i \sin n x \end{array}\right} \quad \text { for } n>0, \ & c_0=\frac{1}{2} a_0, \quad c_n=\frac{1}{2}\left(a_n-i b_n\right), \quad \text { and } \quad c_{-n}=\frac{1}{2}\left(a_n+i b_n\right) \quad \text { for } n>0 . \ & \end{aligned}
Historically the notation with the $a_n$ ‘s and $b_n$ ‘s was introduced first, but the use of complex exponentials has become quite common. Nowadays the notation with $a_n$ ‘s and $b_n$ ‘s tends to be used only when a function $f$ under investigation is real-valued or when all the cosine terms are absent (i.e., $f$ is even) or all the sine terms are absent (i.e., $f$ is odd).
Power series enable us to enlarge our repertory of explicit functions, and the same thing is true of trigonometric series. Just as the coefficients of a power series whose sum is a function $f$ have to be those arising from Taylor’s formula for $f$, the coefficients of a trigonometric series formed from a function have to arise from specific formulas. Let us run through the relevant formal computation: First we observe that the partial sums have to be periodic with period $2 \pi$. The question then is the extent to which a complex-valued periodic function $f$ on the real line can be given by a trigonometric series. Suppose that
$$f(x)=\sum_{n=-\infty}^{\infty} c_n e^{i n x} .$$
Multiply by $e^{-i k x}$ and integrate to get
$$\frac{1}{2 \pi} \int_{-\pi}^\pi f(x) e^{-i k x} d x=\frac{1}{2 \pi} \int_{-\pi}^\pi \sum_{n=-\infty}^{\infty} c_n e^{i n x} e^{-i k x} d x$$

数学代写|实分析代写Real Analysis代考|Weierstrass Approximation Theorem

$$\int f(x-t) \varphi(t) d t$$

数学代写|实分析代写Real Analysis代考|Fourier Series

$$\frac{a_0}{2}+\sum_{n=1}^{\infty}\left(a_n \cos n x+b_n \sin n x\right)$$

\begin{aligned} & \left.\begin{array}{rl} e^{i n x} & =\cos n x+i \sin n x \ e^{-i n x} & =\cos n x-i \sin n x \end{array}\right} \quad \text { for } n>0, \ & c_0=\frac{1}{2} a_0, \quad c_n=\frac{1}{2}\left(a_n-i b_n\right), \quad \text { and } \quad c_{-n}=\frac{1}{2}\left(a_n+i b_n\right) \quad \text { for } n>0 . \ & \end{aligned}

$$f(x)=\sum_{n=-\infty}^{\infty} c_n e^{i n x} .$$

$$\frac{1}{2 \pi} \int_{-\pi}^\pi f(x) e^{-i k x} d x=\frac{1}{2 \pi} \int_{-\pi}^\pi \sum_{n=-\infty}^{\infty} c_n e^{i n x} e^{-i k x} d x$$

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。