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# 数学代写|实分析代写Real Analysis代考|MA507

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## 数学代写|实分析代写Real Analysis代考|Elementary Properties

Although the Fourier transform in the one-variable case dates from the early nineteenth century, it was not until the introduction of the Lebesgue integral early in the twentieth century that the theory could advance very far. Fourier series in one variable have a standard physical interpretation as representing a resolution into component frequencies of a periodic signal that is given as a function of time. In the presence of the Riesz-Fischer Theorem, they are especially handy at analyzing time-independent operators on signals, such as those given by filters. An operator of this kind takes a function $f$ with Fourier series $f(x) \sim \sum_{n=-\infty}^{\infty} c_n e^{i n x}$ into the expression $\sum_{n=-\infty}^{\infty} m_n c_n e^{i n x}$, where the constants $m_n$ depend only on the filter. If the original function $f$ is in $L^2$ and if the constants $m_n$ are bounded, the Riesz-Fischer Theorem allows one to interpret the new series as the Fourier series of a new $L^2$ function $T(f)$, and thus the effect of the filter is to carry $f$ to $T(f)$.

If one imagines that the period is allowed to increase without limit, one can hope to obtain convergence of some sort to a transform that handles aperiodic signals, and this was once a common attitude about how to view the Fourier transform. In the twentieth century the Fourier transform began to be developed as an object in its own right, and soon the theory was extended from one variable to several variables.

The Fourier transform in Euclidean space $\mathbb{R}^N$ is a mapping of suitable kinds of functions on $\mathbb{R}^N$ to other functions on $\mathbb{R}^N$. The functions will in all cases now be assumed to be complex valued. The underlying $\mathbb{R}^N$ is usually regarded as space, rather than time, and the Fourier transform is of great importance in studying operators that commute with translations, i.e., spatially homogeneous operators. One example of such an operator is a linear partial differential operator with constant coefficients, and another is convolution with a fixed function. In the latter case if $\mathcal{F}$ denotes the Fourier transform and $h$ is a fixed function, the relevant formula is $\mathcal{F}(h * f)=\mathcal{F}(h) \mathcal{F}(f)$, the product on the right side being the pointwise product of two functions. Thus convolution can be understood in terms of the simpler operation of pointwise multiplication if we understand what $\mathcal{F}$ does and we understand how to invert $\mathcal{F}$.

## 数学代写|实分析代写Real Analysis代考|Fourier Transform on L1, Inversion Formula

The main theorem of this section is the Fourier inversion formula for $L^1\left(\mathbb{R}^N\right)$. The Fourier transform for $\mathbb{R}^1$ is the analog for the line of the mapping that carries a function $f$ on the circle to its doubly infinite sequence $\left{c_k\right}$ of Fourier coefficients. The inversion problem for the circle amounts to recovering $f$ from the $c_k$ ‘s. We know that the procedure is to form the partial sums $s_n(x)=\sum_{k=-n}^n c_k e^{i k x}$ and to look for a sense in which $\left{s_n\right}$ converges to $f$. There is no problem for the case that $f$ is itself a trigonometric polynomial; then $s_n$ will be equal to $f$ for large enough $n$, and no passage to the limit is necessary.

The situation with the Fourier transform is different. There is no readily available nonzero integrable function on the line analogous to an exponential on the circle for which we know an inversion formula with all constants in place. In order to obtain such an inversion formula for the Fourier transform on $L^1$, it is necessary to be able to invert the Fourier transform of some particular nonzero function explicitly. This step is carried out in Proposition 8.2 below, and then we can address the inversion problem of $L^1\left(\mathbb{R}^N\right)$ in general. The analog for the circle of what we shall prove for the line is a rather modest result: It would say that if $\sum\left|c_k\right|$ is finite, then the sequence of partial sums converges uniformly to a function that equals $f$ almost everywhere. The uniform convergence is a relatively trivial conclusion, being an immediate consequence of the Weierstrass $M$ test; but the conclusion that we recover $f$ lies deeper and incorporates a version of the uniqueness theorem.

## 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。