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

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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
$$\widehat{f}(m)=\int_{\mathbf{T}^n} f(x) e^{-2 \pi i m \cdot x} d x$$

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