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# 数学代写|实分析代写Real Analysis代考|MATH721 Radial symmetry and Bessel functions

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## 数学代写|实分析代写Real Analysis代考|Radial symmetry and Bessel functions

We observed earlier that the Fourier transform of a radial function in $\mathbb{R}^d$ is also radial. In other words, if $f(x)=f_0(|x|)$ for some $f_0$, then $\hat{f}(\xi)=F_0(|\xi|)$ for some $F_0$. A natural problem is to determine a relation between $f_0$ and $F_0$.

This problem has a simple answer in dimensions one and three. If $d=1$ the relation we seek is
$$F_0(\rho)=2 \int_0^{\infty} \cos (2 \pi \rho r) f_0(r) d r$$
If we recall that $\mathbb{R}$ has only two rotations, the identity and multiplication by $-1$, we find that a function is radial precisely when it is even. Having made this observation it is easy to see that if $f$ is radial, and $|\xi|=\rho$, then
\begin{aligned} F_0(\rho)=\hat{f}(|\xi|) & =\int_{-\infty}^{\infty} f(x) e^{-2 \pi i x|\xi|} d x \ & =\int_0^{\infty} f_0(r)\left(e^{-2 \pi i r|\xi|}+e^{2 \pi i r|\xi|}\right) d r \ & =2 \int_0^{\infty} \cos (2 \pi \rho r) f_0(r) d r \end{aligned}

## 数学代写|实分析代写Real Analysis代考|The Radon transform and some of its applications

Invented by Johann Radon in 1917 , the integral transform we discuss next has many applications in mathematics and other sciences, including a significant achievement in medicine. To motivate the definitions and the central problem of reconstruction, we first present the close connection between the Radon transform and the development of $X$-ray scans (or CAT scans) in the theory of medical imaging. The solution of the reconstruction problem, and the introduction of new algorithms and faster computers, all contributed to a rapid development of computerized tomography. In practice, $X$-ray scans provide a “picture” of an internal organ, one that helps to detect and locate many types of abnormalities.

After a brief description of $X$-ray scans in two dimensions, we define the $X$-ray transform and formulate the basic problem of inverting this mapping. Although this problem has an explicit solution in $\mathbb{R}^2$, it is more complicated than the analogous problem in three dimensions, hence we give a complete solution of the reconstruction problem only in $\mathbb{R}^3$. Here we have another example where results are simpler in the odddimensional case than in the even-dimensional situation.

## 数学代写|实分析代写Real Analysis代考|Radial symmetry and Bessel functions

$$F_0(\rho)=2 \int_0^{\infty} \cos (2 \pi \rho r) f_0(r) d r$$

$$F_0(\rho)=\hat{f}(|\xi|)=\int_{-\infty}^{\infty} f(x) e^{-2 \pi i x|\xi|} d x \quad=\int_0^{\infty} f_0(r)\left(e^{-2 \pi i r|\xi|}+e^{2 \pi i r|\xi|}\right) d r=2 \int_0^{\infty} \cos (2 \pi \rho r) f_0(r) d r$$

## 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。