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## 数学代写|傅里叶分析代写Fourier Analysis代考|The Second Key Lemma

Next we need the following lemma.
Lemma 5.5.3. There exists a constant $C=C(n)<\infty$ such that for all $j \geq 1$ and for all $f$ in $L^1\left(\mathbf{R}^n\right)$ we have $$\left|\mathscr{M}j(f)\right|{L^{1, \infty}} \leq C 2^j|f|_{L^1}$$ Proof. Let $K^{(j)}=\left(\varphi_j\right)^{\vee} * d \sigma=\Phi_{2^{-j}} * d \sigma$, where $\Phi$ is a Schwartz function. Setting $$\left(K^{(j)}\right)t(x)=t^{-n} K^{(j)}\left(t^{-1} x\right)$$ we have that $$\mathscr{M}_j(f)=\sup {t>0}\left|\left(K^{(j)}\right)t * f\right|$$ The proof of the lemma is based on the estimate: $$\mathscr{M}_j(f) \leq C 2^j \mathcal{M}(f)$$ and the weak type $(1,1)$ boundedness of the Hardy-Littlewood maximal operator $\mathcal{M}$ (Theorem 2.1.6). To establish (5.5.11), it suffices to show that for any $M>n$ there is a constant $C_M<\infty$ such that $$\left|K^{(j)}(x)\right|=\left|\left(\Phi{2^{-j}} * d \sigma\right)(x)\right| \leq \frac{C_M 2^j}{(1+|x|)^M} .$$
Then Theorem 2.1.10 yields (5.5.11) and hence the required conclusion.
Using the fact that $\Phi$ is a Schwartz function, we have for every $N>0$,
$$\left|\left(\Phi_{2^{-j}} * d \sigma\right)(x)\right| \leq C_N \int_{\mathbf{S}^{n-1}} \frac{2^{n j} d \sigma(y)}{\left(1+2^j|x-y|\right)^N}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Completion of the Proof

It remains to combine the previous ingredients to complete the proof of the theorem. Interpolating between the $L^2 \rightarrow L^2$ and $L^1 \rightarrow L^{1, \infty}$ estimates obtained in Lemmas 5.5.2 and 5.5.3, we obtain
$$\left|\mathscr{M}j(f)\right|{L^p\left(\mathbf{R}^n\right)} \leq C_p 2^{\left(\frac{n}{p}-(n-1)\right) j}|f|_{L^p\left(\mathbf{R}^n\right)}$$
for all $1 \frac{n}{n-1}$ the series $\sum_{j=1}^{\infty} 2^{\left(\frac{n}{p}-(n-1)\right) j}$ converges and we conclude that $\mathscr{M}$ is $L^p$ bounded for these $p$ ‘s. The boundedness of $\mathscr{M}$ on $L^p$ for $p>2$ follows by interpolation between $L^q$ for $q<2$ and the estimate $\mathscr{M}: L^{\infty} \rightarrow L^{\infty}$.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Completion of the Proof

$$\left|\mathscr{M}j(f)\right|{L^p\left(\mathbf{R}^n\right)} \leq C_p 2^{\left(\frac{n}{p}-(n-1)\right) j}|f|_{L^p\left(\mathbf{R}^n\right)}$$

## 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。

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier’s Bold Conjecture

In the early 1800 s Joseph Fourier (along with others) was attempting to mathematically describe the process of heat conduction in a uniform rod of finite length, subject to certain initial and boundary conditions. Fourier’s approach required that the temperature $u(x)$ at position $x$ in the rod at some fixed time be expressed as
\begin{aligned} u(x)= & a_0+a_1 \cos (c x)+b_1 \sin (c x)+a_2 \cos (2 c x)+b_2 \sin (2 c x) \ & +a_3 \cos (3 c x)+b_3 \sin (3 c x)+\cdots \end{aligned}
where $c$ is $\pi$ divided by the rod’s length, and the $a_k$ ‘s and $b_k$ ‘s are constants to be determined after plugging this representation for $u$ into the equations modeling heat flow. (Precisely how they are determined will be discussed in chapter 16, where you will also discover that I’ve simplified things here a bit. For one thing, the $a_k$ ‘s and $b_k$ ‘s are actually functions of time.)
Fourier’s approach was successful, and that idea of representing a function in terms of sines and cosines eventually led to the development of a lot of incredibly useful mathematics.

What Fourier did with the function $u(x)$ was very similar to what we normally do with a three-dimensional vector $\mathbf{v}$. Basically, $\mathbf{v}$ is just some entity possessing “length” and “direction”. Rarely, though, are vector computations done directly using a vector’s length or direction. In practice such computations are normally done using the vector’s components $\left(v_1, v_2, v_3\right)$. For example, the length of $\mathbf{v}$ is usually computed using the component formula
$$|\mathbf{v}|=\sqrt{\left(v_1\right)^2+\left(v_2\right)^2+\left(v_3\right)^2} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier’s Bold Conjecture

Any “reasonable” function can be expressed as a (possibly infinite) linear combination of sines and cosines.

If Fourier’s conjecture is valid, then we should be able to simplify many problems (such as, for example, the problem of mathematically predicting the temperature distribution along a given rod at a given time) by expressing the unknown functions as linear combinations of well-known sine and cosine functions. With luck, the coefficients in these linear combinations will be relatively easy to determine, say, by plugging the expressions into appropriate equations and solving some resulting algebraic equations.

Naturally, it is not all that simple. For one thing, I cannot honestly tell you that Fourier’s conjecture is completely valid, at least not until we better determine what is meant by a function being “reasonable”. But the conjecture turns out to be close enough to the truth to serve as the starting point for our studies, and determining the extent to which this conjecture is valid will be one of our major goals in this text. And, of course, whenever possible, we will want to find out how to compute the “components” of any given (reasonable) function and how to use these components in the manipulations of interest to us (e.g., differentiation, finding solutions to various differential equations, and evaluating functions).

## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier’s Bold Conjecture

\begin{aligned} u(x)= & a_0+a_1 \cos (c x)+b_1 \sin (c x)+a_2 \cos (2 c x)+b_2 \sin (2 c x) \ & +a_3 \cos (3 c x)+b_3 \sin (3 c x)+\cdots \end{aligned}

$$|\mathbf{v}|=\sqrt{\left(v_1\right)^2+\left(v_2\right)^2+\left(v_3\right)^2} .$$

## 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。

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Estimates for Singular Integrals with Rough Kernels

We now turn to another application of the Littlewood-Paley theory involving singular integrals.

Theorem 5.3.4. Suppose that $\mu$ is a finite Borel measure on $\mathbf{R}^n$ with compact support that satisfies $|\widehat{\mu}(\xi)| \leq B \min \left(|\xi|^{-b},|\xi|^b\right)$ for some $b>0$ and all $\xi \neq 0$. Define measures $\mu_j$ by setting $\widehat{\mu_j}(\xi)=\widehat{\mu}\left(2^{-j} \xi\right)$. Then the operator
$$T_\mu(f)(x)=\sum_{j \in \mathbf{Z}}\left(f * \mu_j\right)(x)$$
is bounded on $L^p\left(\mathbf{R}^n\right)$ for all $1<p<\infty$.
Proof. It is natural to begin with the $L^2$ boundedness of $T_\mu$. The estimate on $\widehat{\mu}$ implies that
$$\sum_{j \in \mathbf{Z}}\left|\widehat{\mu}\left(2^{-j} \xi\right)\right| \leq \sum_{j \in \mathbf{Z}} B \min \left(\left|2^{-j} \xi\right|^b,\left|2^{-j} \xi\right|^{-b}\right) \leq C_b B<\infty .$$
The $L^2$ boundedness of $T_\mu$ is an immediate consequence of (5.3.5).

## 数学代写|傅里叶分析代写Fourier Analysis代考|An Almost Orthogonality Principle on $L^p$

Suppose that $T_j$ are multiplier operators given by $T_j(f)=\left(\widehat{f} m_j\right)^{\vee}$, for some multipliers $m_j$. If the functions $m_j$ have disjoint supports and they are bounded uniformly in $j$, then the operator
$$T=\sum_j T_j$$
is bounded on $L^2$. The following theorem gives an $L^p$ analogue of this result.
Theorem 5.3.6. Suppose that $1<p \leq 2 \leq q<\infty$. Let $m_j$ be Schwartz functions supported in the annuli $2^{j-1} \leq|\xi| \leq 2^{j+1}$ and let $T_j(f)=\left(\widehat{f} m_j\right)^{\vee}$. Suppose that the $T_j$ ‘s are uniformly bounded operators from $L^p\left(\mathbf{R}^n\right)$ to $L^q\left(\mathbf{R}^n\right)$, i.e.,
$$\sup j\left|T_j\right|{L^p \rightarrow L^q}=A<\infty .$$
Then for each $f \in L^p\left(\mathbf{R}^n\right)$, the series
$$T(f)=\sum_j T_j(f)$$
converges in the $L^q$ norm and there exists a constant $C_{p, q, n}<\infty$ such that
$$|T|_{L^p \rightarrow L^q} \leq C_{p, q, n} A .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Estimates for Singular Integrals with Rough Kernels

$$T_\mu(f)(x)=\sum_{j \in \mathbf{Z}}\left(f * \mu_j\right)(x)$$

$$\sum_{j \in \mathbf{Z}}\left|\widehat{\mu}\left(2^{-j} \xi\right)\right| \leq \sum_{j \in \mathbf{Z}} B \min \left(\left|2^{-j} \xi\right|^b,\left|2^{-j} \xi\right|^{-b}\right) \leq C_b B<\infty .$$
$T_\mu$的$L^2$有界性是式(5.3.5)的直接结果。

## 数学代写|傅里叶分析代写Fourier Analysis代考|An Almost Orthogonality Principle on $L^p$

$$T=\sum_j T_j$$

$$\sup j\left|T_j\right|{L^p \rightarrow L^q}=A<\infty .$$

$$T(f)=\sum_j T_j(f)$$

$$|T|{L^p \rightarrow L^q} \leq C{p, q, n} A .$$

## 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。

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Applications and $\ell^r$-Valued Extensions of Linear Operators

Here is an application of Theorem 4.5.1:
Example 4.5.3. On the real line consider the intervals $I_j=\left[b_j, \infty\right)$ for $j \in \mathbf{Z}$. Let $T_j$ be the operator given by multiplication on the Fourier transform by the characteristic function of $I_j$. Then we have the following two inequalities:

$$\begin{gathered} \left|\left(\sum_{j \in \mathbf{Z}}\left|T_j\left(f_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^p} \leq C_p\left|\left(\sum_{j \in \mathbf{Z}}\left|f_j\right|^2\right)^{\frac{1}{2}}\right|_{L^p}, \ \left|\left(\sum_{j \in \mathbf{Z}}\left|T_j\left(f_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^{1, \infty}} \leq C\left|\left(\sum_{j \in \mathbf{Z}}\left|f_j\right|^2\right)^{\frac{1}{2}}\right|_{L^1}, \end{gathered}$$
for $1<p<\infty$. To prove these, first observe that the operator $T=\frac{1}{2}(I+i H)$ is given on the Fourier transform by multiplication by the characteristic function of the halfaxis $[0, \infty)$ [precisely, the Fourier multiplier of $T$ is equal to 1 on the set $(0, \infty)$ and $1 / 2$ at the origin; this function is almost everywhere equal to the characteristic function of the half-axis $[0, \infty)]$. Moreover, each $T_j$ is given by
$$T_j(f)(x)=e^{2 \pi i b_j x} T\left(e^{-2 \pi i b_j(\cdot)} f\right)(x)$$
and thus with $g_j(x)=e^{-2 \pi i b_j x} f(x)$, (4.5.13) and (4.5.14) can be written respectively as
$$\begin{gathered} \left|\left(\sum_{j \in \mathbf{Z}}\left|T\left(g_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^p} \leq C_p\left|\left(\sum_{j \in \mathbf{Z}}\left|g_j\right|^2\right)^{\frac{1}{2}}\right|_{L^p}, \ \left|\left(\sum_{j \in \mathbf{Z}}\left|T\left(g_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^{1, \infty}} \leq C\left|\left(\sum_{j \in \mathbf{Z}}\left|g_j\right|^2\right)^{\frac{1}{2}}\right|_{L^1} \end{gathered}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|General Banach-Valued Extensions

We now set up the background required to state the main results of this section. Although the Banach spaces of most interest to us are $\ell^r$ for $1 \leq r \leq \infty$, we introduce the basic notions we need in general.

Let $\mathscr{B}$ be a Banach space over the field of complex numbers with norm ||$_{\mathscr{B}}$, and let $\mathscr{B}^$ be its dual (with norm ||$_{\mathscr{B}^}$ ). A function $F$ defined on a $\sigma$-finite measure space $(X, \mu)$ and taking values in $\mathscr{B}$ is called $\mathscr{B}$-measurable if there exists a measurable subset $X_0$ of $X$ such that $\mu\left(X \backslash X_0\right)=0, F\left[X_0\right]$ is contained in some separable subspace $\mathscr{B}0$ of $\mathscr{B}$, and for every $u^* \in \mathscr{B}^$ the complex-valued map $$x \mapsto\left\langle u^, F(x)\right\rangle$$
is measurable. A consequence of this definition is that the positive function $x \mapsto$ $|F(x)|{\mathscr{B}}$ on $X$ is measurable; to see this, use the relevant result in Yosida [296, p. 131].

For $0<p \leq \infty$, denote by $L^p(X, \mathscr{B})$ the space of all $\mathscr{B}$-measurable functions $F$ on $X$ satisfying
$$\left(\int_X|F(x)|_{\mathscr{B}}^p d \mu(x)\right)^{\frac{1}{p}}<\infty$$
with the obvious modification when $p=\infty$. Similarly define $L^{p, \infty}(X, \mathscr{B})$ as the space of all $\mathscr{B}$-measurable functions $F$ on $X$ satisfying
$$|| F(\cdot)\left|_{\mathscr{B}}\right|_{L^{p, \infty}(X)}<\infty$$
Then $L^p(X, \mathscr{B})$ (respectively, $L^{p, \infty}(X, \mathscr{B})$ ) is called the $L^p$ (respectively, $L^{p, \infty}$ ) space of functions on $X$ with values in $\mathscr{B}$. Similarly, we can define other Lorentz spaces of $\mathscr{B}$-valued functions. The quantity in (4.5.17) (respectively, in (4.5.18)) is the norm of $F$ in $L^p(X, \mathscr{B})$ (respectively, in $L^{p, \infty}(X, \mathscr{B})$ ).

We denote by $L^p(X)$ the space $L^p(X, \mathbf{C})$. Let $L^p(X) \otimes \mathscr{B}$ be the set of all finite linear combinations of elements of $\mathscr{B}$ with coefficients in $L^p(X)$, that is, elements of the form
$$F=f_1 u_1+\cdots+f_m u_m$$
where $f_j \in L^p(X), u_j \in \mathscr{B}$, and $m \in \mathbf{Z}^{+}$.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Applications and $\ell^r$-Valued Extensions of Linear Operators

$$\begin{gathered} \left|\left(\sum_{j \in \mathbf{Z}}\left|T_j\left(f_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^p} \leq C_p\left|\left(\sum_{j \in \mathbf{Z}}\left|f_j\right|^2\right)^{\frac{1}{2}}\right|_{L^p}, \ \left|\left(\sum_{j \in \mathbf{Z}}\left|T_j\left(f_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^{1, \infty}} \leq C\left|\left(\sum_{j \in \mathbf{Z}}\left|f_j\right|^2\right)^{\frac{1}{2}}\right|_{L^1}, \end{gathered}$$
for $1<p<\infty$. To prove these, first observe that the operator $T=\frac{1}{2}(I+i H)$ is given on the Fourier transform by multiplication by the characteristic function of the halfaxis $[0, \infty)$ [precisely, the Fourier multiplier of $T$ is equal to 1 on the set $(0, \infty)$ and $1 / 2$ at the origin; this function is almost everywhere equal to the characteristic function of the half-axis $[0, \infty)]$. Moreover, each $T_j$ is given by
$$T_j(f)(x)=e^{2 \pi i b_j x} T\left(e^{-2 \pi i b_j(\cdot)} f\right)(x)$$
and thus with $g_j(x)=e^{-2 \pi i b_j x} f(x)$, (4.5.13) and (4.5.14) can be written respectively as
$$\begin{gathered} \left|\left(\sum_{j \in \mathbf{Z}}\left|T\left(g_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^p} \leq C_p\left|\left(\sum_{j \in \mathbf{Z}}\left|g_j\right|^2\right)^{\frac{1}{2}}\right|_{L^p}, \ \left|\left(\sum_{j \in \mathbf{Z}}\left|T\left(g_j\right)\right|^2\right)^{\frac{1}{2}}\right|_{L^{1, \infty}} \leq C\left|\left(\sum_{j \in \mathbf{Z}}\left|g_j\right|^2\right)^{\frac{1}{2}}\right|_{L^1} \end{gathered}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|General Banach-Valued Extensions

$$\left(\int_X|F(x)|{\mathscr{B}}^p d \mu(x)\right)^{\frac{1}{p}}<\infty$$ 与明显的修改时$p=\infty$。同样地，将$L^{p, \infty}(X, \mathscr{B})$定义为$X$上满足的所有$\mathscr{B}$ -可测量函数$F$的空间 $$|| F(\cdot)\left|{\mathscr{B}}\right|_{L^{p, \infty}(X)}<\infty$$

$$F=f_1 u_1+\cdots+f_m u_m$$

## 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。

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Sufficient Conditions for $L^p$ Boundedness of Singular Integrals

We first note that under conditions (4.4.1), (4.4.2), and (4.4.3), there exists a tempered distribution $W$ that coincides with $K$ on $\mathbf{R}^n \backslash{0}$. Indeed, condition (4.4.3) implies that there exists a sequence $\delta_j \downarrow 0$ such that
$$\lim {j \rightarrow \infty} \int{\delta_j<|x| \leq 1} K(x) d x=L$$

exists. Using (4.3.8), we conclude that there exists such a tempered distribution $W$. Note that we must have $|L| \leq A_3$.
We observe that the difference of two distributions $W$ and $W^{\prime}$ that coincide with $K$ on $\mathbf{R}^n \backslash{0}$ must be supported at the origin.

Theorem 4.4.1. Assume that $K$ satisfies (4.4.1), (4.4.2), and (4.4.3), and let $W$ be a tempered distribution that coincides with $K$ on $\mathbf{R}^n \backslash{0}$. Then we have
$$\sup {0<\varepsilon{\xi \neq 0}\left|\left(K \chi_{\varepsilon<|\cdot|<N}\right)(\xi)\right| \leq 15\left(A_1+A_2+A_3\right) \text {. }$$
Thus the operator given by convolution with $W$ maps $L^2\left(\mathbf{R}^n\right)$ to itself with norm at most $15\left(A_1+A_2+A_3\right)$. Consequently, it also maps $L^1\left(\mathbf{R}^n\right)$ to $L^{1, \infty}\left(\mathbf{R}^n\right)$ with bound at most a dimensional constant multiple of $A_1+A_2+A_3$ and $L^p\left(\mathbf{R}^n\right)$ to itself with bound at most $C_n \max \left(p,(p-1)^{-1}\right)\left(A_1+A_2+A_3\right)$, for some dimensional constant $C_n$, whenever $1<p<\infty$.

Proof. Let us set $K^{(\varepsilon, N)}(x)=K(x) \chi_{\varepsilon<|x|<N}$. If we prove (4.4.4), then for all $f$ in $\mathscr{S}\left(\mathbf{R}^n\right)$ we will have the estimate
$$\left|f * K^{\left(\delta_j, j\right)}\right|_{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|_{L^2}$$
uniformly in $j$. Using this, (4.3.9), and Fatou’s lemma, we obtain that
$$|f * W|_{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|_{L^2},$$
thus proving the second conclusion of the theorem.
Let us now fix a $\xi$ with $\varepsilon<|\xi|^{-1}<N$ and prove (4.4.4). Write $\widehat{K^{(\varepsilon, N)}}(\xi)=$ $I_1(\xi)+I_2(\xi)$, where
\begin{aligned} & I_1(\xi)=\int_{\varepsilon<|x|<|\xi|^{-1}} K(x) e^{-2 \pi i x \cdot \xi} d x \ & I_2(\xi)=\int_{|\xi|^{-1}<|x|<N} K(x) e^{-2 \pi i x \cdot \xi} d x \end{aligned}

## 数学代写|傅里叶分析代写Fourier Analysis代考|An Example

We now give an example of a distribution that satisfies conditions (4.4.1), (4.4.2), and (4.4.3).

Example 4.4.2. Let $\tau$ be a nonzero real number and let $K(x)=\frac{1}{\mid x^{n+i \tau}}$ defined for $x \in \mathbf{R}^n \backslash{0}$. For a sequence $\delta_k \downarrow 0$ and $\varphi$ a Schwartz function on $\mathbf{R}^n$, define
$$\langle W, \varphi\rangle=\lim {k \rightarrow \infty} \int{\delta_k \leq|x|} \varphi(x) \frac{d x}{|x|^{n+i \tau}},$$
whenever the limit exists. We claim that for some choices of sequences $\delta_k, W$ is a well defined tempered distribution on $\mathbf{R}^n$. Take, for example, $\delta_k=e^{-2 \pi k / \tau}$. For this sequence $\delta_k$, observe that
$$\int_{\delta_k \leq|x| \leq 1} \frac{1}{|x|^{n+i \tau}} d x=\omega_{n-1} \frac{1-\left(e^{-2 \pi k / \tau}\right)^{-i \tau}}{-i \tau}=0,$$
and thus
$$\langle W, \varphi\rangle=\int_{|x| \leq 1}(\varphi(x)-\varphi(0)) \frac{d x}{|x|^{n+i \tau}}+\int_{|x| \geq 1} \varphi(x) \frac{d x}{|x|^{n+i \tau}},$$
which implies that $W \in \mathscr{S}^{\prime}\left(\mathbf{R}^n\right)$, since
$$|\langle W, \varphi\rangle| \leq C\left[|\nabla \varphi|_{L^{\infty}}+||x| \varphi(x)|_{L^{\infty}}\right]$$
If $\varphi$ is supported in $\mathbf{R}^n \backslash{0}$, then
$$\langle W, \varphi\rangle=\int K(x) \varphi(x) d x$$
Therefore $W$ coincides with the function $K$ away from the origin. Moreover, (4.4.1) and (4.4.2) are clearly satisfied for $K$, while (4.4.3) is also satisfied, since
$$\left|\int_{R_1<|x|<R_2} \frac{1}{|x|^{n+i \tau}} d x\right|=\omega_{n-1}\left|\frac{R_1^{-i \tau}-R_2^{-i \tau}}{-i \tau}\right| \leq \frac{2 \omega_{n-1}}{|\tau|}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Sufficient Conditions for $L^p$ Boundedness of Singular Integrals

$$\lim {j \rightarrow \infty} \int{\delta_j<|x| \leq 1} K(x) d x=L$$

$$\sup {0<\varepsilon{\xi \neq 0}\left|\left(K \chi_{\varepsilon<|\cdot|<N}\right)(\xi)\right| \leq 15\left(A_1+A_2+A_3\right) \text {. }$$

$$\left|f * K^{\left(\delta_j, j\right)}\right|{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|{L^2}$$

$$|f * W|{L^2} \leq 15\left(A_1+A_2+A_3\right)|f|{L^2},$$

$$\left{\lambda_{k r+s}: k \in \mathbf{Z}^{+} \cup{0}\right}=\left{\mu_1, \mu_2, \mu_3, \ldots\right}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Definition and Basic Properties of the Hilbert Transform

$$\left\langle W_0, \varphi\right\rangle=\frac{1}{\pi} \lim {\varepsilon \rightarrow 0} \int{\varepsilon \leq|x| \leq 1} \frac{\varphi(x)}{x} d x+\frac{1}{\pi} \int_{|x| \geq 1} \frac{\varphi(x)}{x} d x$$

$f \in \mathscr{S}(\mathbf{R})$的希尔伯特变换定义为
$$H(f)(x)=\left(W_0 * f\right)(x)=\lim _{\varepsilon \rightarrow 0} H^{(\varepsilon)}(f)(x)$$

## 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。

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

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Equivalent Formulations of Convergence in Norm

The question we pose is for which $1 \leq p<\infty$ we have
$$|D(n, N) * f-f|_{L^p\left(\mathbf{T}^n\right)} \rightarrow 0 \quad \text { as } N \rightarrow \infty$$
and similarly for the circular Dirichlet kernel $\widetilde{D}(n, N)$. We tackle this question by looking at an equivalent formulation of it.
Theorem 3.5.1. Fix $1 \leq p<\infty$ and $\left{a_m\right}$ in $\ell^{\infty}\left(\mathbf{Z}^n\right)$. For each $R \geq 0$, let $\left{a_m(R)\right}_{m \in \mathbf{Z}^n}$ be a compactly supported sequence (whose support depends on $R$ ) that satisfies $\lim {R \rightarrow \infty} a_m(R)=a_m$. For $f \in L^p\left(\mathbf{T}^n\right)$ define $$S_R(f)(x)=\sum{m \in \mathbf{Z}^n} a_m(R) \widehat{f}(m) e^{2 \pi i m \cdot x}$$
and for $h \in \mathscr{C}^{\infty}\left(\mathbf{T}^n\right)$ define
$$A(h)(x)=\sum_{m \in \mathbf{Z}^n} a_m \widehat{h}(m) e^{2 \pi i m \cdot x}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Functions with Absolutely Summable Fourier Coefficients

Decay for the Fourier coefficients can also be indirectly deduced from a certain knowledge about the summability of these coefficients. The simplest such kind of summability is in the sense of $\ell^1$. It is therefore natural to consider the class of functions on the torus whose Fourier coefficients form an absolutely summable series.
Definition 3.2.15. An integrable function $f$ on the torus is said to have an absolutely convergent Fourier series if
$$\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|<+\infty$$
We denote by $A\left(\mathbf{T}^n\right)$ the space of all integrable functions on the torus $\mathbf{T}^n$ whose Fourier series are absolutely convergent. We then introduce a norm on $A\left(\mathbf{T}^n\right)$ by setting
$$|f|_{A\left(\mathbf{T}^n\right)}=\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|$$
It is straightforward that every function in $A\left(\mathbf{T}^n\right)$ must be bounded. The following theorem gives us a sufficient condition for a function to be in $A\left(\mathbf{T}^n\right)$.

Theorem 3.2.16. Let $s$ be a nonnegative integer and let $0 \leq \alpha<1$. Assume that $f$ is a function defined on $\mathbf{T}^n$ all of whose partial derivatives of order $s$ lie in the space $\dot{\Lambda}\alpha$. Suppose that $s+\alpha>n / 2$. Then $f \in A\left(\mathbf{T}^n\right)$ and $$|f|{A\left(\mathbf{T}^n\right)} \leq C \sup {|\beta|=s}\left|\partial^\beta f\right|{\dot{\Lambda}_\alpha}$$
where $C$ depends on $n, \alpha$, and $s$.
Proof. For $1 \leq j \leq n$, let $e_j$ be the element of $\mathbf{R}^n$ with zero entries except for the $j$ th coordinate, which is 1 . Let $l$ be a positive integer and let $h_j=2^{-l-2} e_j$.

Then for a multi-index $m=\left(m_1, \ldots, m_n\right)$ satisfying $2^l \leq|m| \leq 2^{l+1}$ and for $j$ in ${1, \ldots, n}$ chosen such that $\left|m_j\right|=\sup _k\left|m_k\right|$ we have
$$\frac{\left|m_j\right|}{2^l} \geq \frac{|m|}{2^l \sqrt{n}} \geq \frac{1}{\sqrt{n}}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Decay of Fourier Coefficients of Smooth Functions

3.2.5.定义对于$0 \leq \gamma<1$
$$|f|{\dot{\Lambda}\gamma}=\sup {x, h \in \mathbf{T}^n} \frac{|f(x+h)-f(x)|}{|h|^\gamma}$$ 和 $$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|{\dot{\lambda}\gamma}<\infty\right}$$我们称$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$为环面上阶为$\gamma$的齐次Lipschitz空间。在$\mathbf{T}^n$上与$|f|{\dot{\Lambda}\gamma}<\infty$的函数$f$称为$\gamma$阶的齐次Lipschitz函数。有些话是适当的。3.2.6.备注$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$在$\mathbf{T}^n$上被称为$\gamma$阶的齐次Lipschitz空间，而$\Lambda_\gamma\left(\mathbf{T}^n\right)$则被称为$\gamma$阶的Lipschitz空间。后一个空间定义为
$$\Lambda_\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|{\Lambda\gamma}<\infty\right},$$

$$|f|{\Lambda\gamma}=|f|{L^{\infty}}+|f|{\dot{\Lambda}_\gamma} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Functions with Absolutely Summable Fourier Coefficients

3.2.15.定义环面上的可积函数$f$有绝对收敛的傅立叶级数，如果
$$\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|<+\infty$$

$$|f|{A\left(\mathbf{T}^n\right)}=\sum{m \in \mathbf{Z}^n}|\widehat{f}(m)|$$

$$\frac{\left|m_j\right|}{2^l} \geq \frac{|m|}{2^l \sqrt{n}} \geq \frac{1}{\sqrt{n}}$$

## 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。

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

## avatest™帮您通过考试

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## 数学代写|傅里叶分析代写Fourier Analysis代考|Decay of Fourier Coefficients of Smooth Functions

We next study the decay of the Fourier coefficients of functions that possess a certain amount of smoothness. In this section we see that the decay of the Fourier coefficients reflects the smoothness of the function in a rather precise quantitative way. Conversely, if the Fourier coefficients of an integrable function have polynomial decay faster than the dimension, then a certain amount of smoothness can be inferred about the function.
Definition 3.2.5. For $0 \leq \gamma<1$ define
$$|f|_{\dot{\Lambda}\gamma}=\sup {x, h \in \mathbf{T}^n} \frac{|f(x+h)-f(x)|}{|h|^\gamma}$$
and
$$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|{\dot{\lambda}\gamma}<\infty\right}$$ We call $\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$ the homogeneous Lipschitz space of order $\gamma$ on the torus. Functions $f$ on $\mathbf{T}^n$ with $|f|_{\dot{\Lambda}\gamma}<\infty$ are called homogeneous Lipschitz functions of order $\gamma$. Some remarks are in order. Remark 3.2.6. $\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$ is called the homogeneous Lipschitz space of order $\gamma$ on $\mathbf{T}^n$, in contrast to the space $\Lambda_\gamma\left(\mathbf{T}^n\right)$, which is called the Lipschitz space of order $\gamma$. The latter space is defined as
$$\Lambda_\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|_{\Lambda_\gamma}<\infty\right},$$
where
$$|f|_{\Lambda_\gamma}=|f|_{L^{\infty}}+|f|_{\dot{\Lambda}_\gamma} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Functions with Absolutely Summable Fourier Coefficients

Decay for the Fourier coefficients can also be indirectly deduced from a certain knowledge about the summability of these coefficients. The simplest such kind of summability is in the sense of $\ell^1$. It is therefore natural to consider the class of functions on the torus whose Fourier coefficients form an absolutely summable series.
Definition 3.2.15. An integrable function $f$ on the torus is said to have an absolutely convergent Fourier series if
$$\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|<+\infty$$
We denote by $A\left(\mathbf{T}^n\right)$ the space of all integrable functions on the torus $\mathbf{T}^n$ whose Fourier series are absolutely convergent. We then introduce a norm on $A\left(\mathbf{T}^n\right)$ by setting
$$|f|_{A\left(\mathbf{T}^n\right)}=\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|$$
It is straightforward that every function in $A\left(\mathbf{T}^n\right)$ must be bounded. The following theorem gives us a sufficient condition for a function to be in $A\left(\mathbf{T}^n\right)$.

Theorem 3.2.16. Let $s$ be a nonnegative integer and let $0 \leq \alpha<1$. Assume that $f$ is a function defined on $\mathbf{T}^n$ all of whose partial derivatives of order $s$ lie in the space $\dot{\Lambda}\alpha$. Suppose that $s+\alpha>n / 2$. Then $f \in A\left(\mathbf{T}^n\right)$ and $$|f|{A\left(\mathbf{T}^n\right)} \leq C \sup {|\beta|=s}\left|\partial^\beta f\right|{\dot{\Lambda}_\alpha}$$
where $C$ depends on $n, \alpha$, and $s$.
Proof. For $1 \leq j \leq n$, let $e_j$ be the element of $\mathbf{R}^n$ with zero entries except for the $j$ th coordinate, which is 1 . Let $l$ be a positive integer and let $h_j=2^{-l-2} e_j$.

Then for a multi-index $m=\left(m_1, \ldots, m_n\right)$ satisfying $2^l \leq|m| \leq 2^{l+1}$ and for $j$ in ${1, \ldots, n}$ chosen such that $\left|m_j\right|=\sup _k\left|m_k\right|$ we have
$$\frac{\left|m_j\right|}{2^l} \geq \frac{|m|}{2^l \sqrt{n}} \geq \frac{1}{\sqrt{n}}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Decay of Fourier Coefficients of Smooth Functions

3.2.5.定义对于$0 \leq \gamma<1$
$$|f|{\dot{\Lambda}\gamma}=\sup {x, h \in \mathbf{T}^n} \frac{|f(x+h)-f(x)|}{|h|^\gamma}$$ 和 $$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|{\dot{\lambda}\gamma}<\infty\right}$$我们称$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$为环面上阶为$\gamma$的齐次Lipschitz空间。在$\mathbf{T}^n$上与$|f|{\dot{\Lambda}\gamma}<\infty$的函数$f$称为$\gamma$阶的齐次Lipschitz函数。有些话是适当的。3.2.6.备注$\dot{\Lambda}\gamma\left(\mathbf{T}^n\right)$在$\mathbf{T}^n$上被称为$\gamma$阶的齐次Lipschitz空间，而$\Lambda_\gamma\left(\mathbf{T}^n\right)$则被称为$\gamma$阶的Lipschitz空间。后一个空间定义为
$$\Lambda_\gamma\left(\mathbf{T}^n\right)=\left{f: \mathbf{T}^n \rightarrow \mathbf{C} \text { with }|f|{\Lambda\gamma}<\infty\right},$$

$$|f|{\Lambda\gamma}=|f|{L^{\infty}}+|f|{\dot{\Lambda}_\gamma} .$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Functions with Absolutely Summable Fourier Coefficients

3.2.15.定义环面上的可积函数$f$有绝对收敛的傅立叶级数，如果
$$\sum_{m \in \mathbf{Z}^n}|\widehat{f}(m)|<+\infty$$

$$|f|{A\left(\mathbf{T}^n\right)}=\sum{m \in \mathbf{Z}^n}|\widehat{f}(m)|$$

$$\frac{\left|m_j\right|}{2^l} \geq \frac{|m|}{2^l \sqrt{n}} \geq \frac{1}{\sqrt{n}}$$

## 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。

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

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## 数学代写|傅里叶分析代写Fourier Analysis代考|The n-Torus $\mathbf{T}^n$

The $n$-torus $\mathbf{T}^n$ is the cube $[0,1]^n$ with opposite sides identified. This means that the points $\left(x_1, \ldots, 0, \ldots, x_n\right)$ and $\left(x_1, \ldots, 1, \ldots, x_n\right)$ are identified whenever 0 and 1 appear in the same coordinate. A more precise definition can be given as follows: For $x, y$ in $\mathbf{R}^n$, we say that
$$x \equiv y$$
if $x-y \in \mathbf{Z}^n$. Here $\mathbf{Z}^n$ is the additive subgroup of all points in $\mathbf{R}^n$ with integer coordinates. If (3.1.1) holds, then we write $x=y(\bmod 1)$. It is a simple fact that $\equiv$ is an equivalence relation that partitions $\mathbf{R}^n$ into equivalence classes. The $n$-torus $\mathbf{T}^n$ is then defined as the set $\mathbf{R}^n / \mathbf{Z}^n$ of all such equivalence classes. When $n=1$, this set can be geometrically viewed as a circle by bending the line segment $[0,1]$ so that its endpoints are brought together. When $n=2$, the identification brings together the left and right sides of the unit square $[0,1]^2$ and then the top and bottom sides as well. The resulting figure is a two-dimensional manifold embedded in $\mathbf{R}^3$ that looks like a donut. See Figure 3.1 .

The $n$-torus is an additive group, and zero is the identity element of the group, which of course coincides with every $e_j=(0, \ldots, 0,1,0, \ldots, 0)$. To avoid multiple appearances of the identity element in the group, we often think of the $n$-torus as the set $[-1 / 2,1 / 2]^n$. Since the group $\mathbf{T}^n$ is additive, the inverse of an element $x \in \mathbf{T}^n$ is denoted by $-x$. For example, $-(1 / 3,1 / 4) \equiv(2 / 3,3 / 4)$ on $\mathbf{T}^2$, or, equivalently, $-(1 / 3,1 / 4)=(2 / 3,3 / 4)(\bmod 1)$
The $n$-torus $\mathbf{T}^n$ can also be thought of as the following subset of $\mathbf{C}^n$,
$$\left{\left(e^{2 \pi i x_1}, \ldots, e^{2 \pi i x_n}\right) \in \mathbf{C}^n:\left(x_1, \ldots, x_n\right) \in[0,1]^n\right}$$
in a way analogous to which the unit interval $[0,1]$ can be thought of as the unit circle in $\mathbf{C}$ once 1 and 0 are identified.

## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier Coefficients

Definition 3.1.1. For a complex-valued function $f$ in $L^1\left(\mathbf{T}^n\right)$ and $m$ in $\mathbf{Z}^n$, we define
$$\widehat{f}(m)=\int_{\mathbf{T}^n} f(x) e^{-2 \pi i m \cdot x} d x$$
We call $\widehat{f}(m)$ the $m$ th Fourier coefficient of $f$. We note that $\widehat{f}(m)$ is not defined for $\xi \in \mathbf{R}^n \backslash \mathbf{Z}^n$, since the function $x \mapsto e^{-2 \pi i \xi \cdot x}$ is not 1-periodic in every coordinate and therefore not well defined on $\mathbf{T}^n$.
The Fourier series of $f$ at $x \in \mathbf{T}^n$ is the series
$$\sum_{m \in \mathbf{Z}^n} \widehat{f}(m) e^{2 \pi i m \cdot x} .$$
It is not clear at present in which sense and for which $x \in \mathbf{T}^n$ (3.1.5) converges. The study of convergence of Fourier series is the main topic of study in this chapter.
We quickly recall the notation we introduced in Chapter 2 . We denote by $\bar{f}$ the complex conjugate of the function $f$, by $\tilde{f}$ the function $\tilde{f}(x)=f(-x)$, and by $\tau^y(f)$ the function $\tau^y(f)(x)=f(x-y)$ for all $y \in \mathbf{T}^n$. We mention some elementary properties of Fourier coefficients.

## 数学代写|傅里叶分析代写Fourier Analysis代考|The n-Torus $\mathbf{T}^n$

$n$ -环面$\mathbf{T}^n$是对边相等的立方体$[0,1]^n$。这意味着当0和1出现在同一坐标中时，点$\left(x_1, \ldots, 0, \ldots, x_n\right)$和$\left(x_1, \ldots, 1, \ldots, x_n\right)$就会被识别。更精确的定义如下:对于$\mathbf{R}^n$中的$x, y$，我们说
$$x \equiv y$$

$n$ -环面是一个加性群，0是这个群的单位元，它当然与每个$e_j=(0, \ldots, 0,1,0, \ldots, 0)$重合。为了避免标识元素在组中多次出现，我们通常将$n$ -环面视为集合$[-1 / 2,1 / 2]^n$。因为组$\mathbf{T}^n$是可加的，所以元素$x \in \mathbf{T}^n$的逆表示为$-x$。例如，$\mathbf{T}^2$上的$-(1 / 3,1 / 4) \equiv(2 / 3,3 / 4)$，或$-(1 / 3,1 / 4)=(2 / 3,3 / 4)(\bmod 1)$
$n$ -环面$\mathbf{T}^n$也可以被认为是$\mathbf{C}^n$的以下子集，
$$\left{\left(e^{2 \pi i x_1}, \ldots, e^{2 \pi i x_n}\right) \in \mathbf{C}^n:\left(x_1, \ldots, x_n\right) \in[0,1]^n\right}$$

## 数学代写|傅里叶分析代写Fourier Analysis代考|Fourier Coefficients

3.1.1.定义对于$L^1\left(\mathbf{T}^n\right)$中的复值函数$f$和$\mathbf{Z}^n$中的$m$，我们定义
$$\widehat{f}(m)=\int_{\mathbf{T}^n} f(x) e^{-2 \pi i m \cdot x} d x$$

## 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。