Posted on Categories:Measure Theory and Fourier Analysis, 傅里叶分析, 数学代写, 测度论和傅里叶分析

# 数学代写|傅里叶分析代写Fourier Analysis代考|Complex Method: The Riesz-Thorin Interpolation Theorem

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|傅里叶分析代写Fourier Analysis代考|Complex Method: The Riesz-Thorin Interpolation Theorem

The next interpolation theorem assumes stronger endpoint estimates, but yields a more natural bound on the norm of the operator on the intermediate spaces. Unfortunately, it is mostly applicable for linear operators and in some cases for sublinear operators (often via a linearization process). It does not apply to quasilinear operators without some loss in the constant. A short history of this theorem is discussed at the end of this chapter.

Theorem 1.3.4. Let $(X, \mu)$ and $(Y, v)$ be two measure spaces. Let $T$ be a linear operator defined on the set of all simple functions on $X$ and taking values in the set of measurable functions on $Y$. Let $1 \leq p_0, p_1, q_0, q_1 \leq \infty$ and assume that
\begin{aligned} & |T(f)|_{L^{q_0}} \leq M_0|f|_{L^{p_0}}, \ & |T(f)|_{L^{q_1}} \leq M_1|f|_{L^{p_1}}, \end{aligned}
for all simple functions $f$ on $X$. Then for all $0<\theta<1$ we have
$$|T(f)|_{L^q} \leq M_0^{1-\theta} M_1^\theta|f|_{L^p}$$

for all simple functions $f$ on $X$, where
$$\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1} \quad \text { and } \quad \frac{1}{q}=\frac{1-\theta}{q_0}+\frac{\theta}{q_1} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Interpolation of Analytic Families of Operators

Theorem 1.3.4 can now be extended to the case in which the interpolated operators are allowed to vary. In particular, if a family of operators depends analytically on a parameter $z$, then the proof of this theorem can be adapted to work in this setting.
We now describe the setup for this theorem. Let $(X, \mu)$ and $(Y, v)$ be measure spaces. Suppose that for every $z$ in the closed strip $\bar{S}={z \in \mathbf{C}: 0 \leq \operatorname{Re} z \leq 1}$ there is an associated linear operator $T_z$ defined on the space of simple functions on $X$ and taking values in the space of measurable functions on $Y$ such that
$$\int_Y\left|T_z(f) g\right| d v<\infty$$
whenever $f$ and $g$ are simple functions on $X$ and $Y$, respectively. The family $\left{T_z\right}_z$ is said to be analytic if the function
$$z \mapsto \int_Y T_z(f) g d v$$
is analytic in the open strip $S={z \in \mathbf{C}: 0<\operatorname{Re} z<1}$ and continuous on its closure. Finally, the analytic family is of admissible growth if there is a constant $a<\pi$ and a constant $C_{f, g}$ such that
$$e^{-a|\operatorname{Im} z|} \log \left|\int_Y T_z(f) g d v\right| \leq C_{f, g}<\infty$$
for all $z$ satisfying $0 \leq \operatorname{Re} z \leq 1$. The extension of the Riesz-Thorin interpolation theorem is now stated.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Complex Method: The Riesz-Thorin Interpolation Theorem

\begin{aligned} & |T(f)|{L^{q_0}} \leq M_0|f|{L^{p_0}}, \ & |T(f)|{L^{q_1}} \leq M_1|f|{L^{p_1}}, \end{aligned}

$$|T(f)|{L^q} \leq M_0^{1-\theta} M_1^\theta|f|{L^p}$$

$$\frac{1}{p}=\frac{1-\theta}{p_0}+\frac{\theta}{p_1} \quad \text { and } \quad \frac{1}{q}=\frac{1-\theta}{q_0}+\frac{\theta}{q_1} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Interpolation of Analytic Families of Operators

$$\int_Y\left|T_z(f) g\right| d v<\infty$$

$$e^{-a|\operatorname{Im} z|} \log \left|\int_Y T_z(f) g d v\right| \leq C_{f, g}<\infty$$

## 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。