### 数学代写|偏微分方程代写partial difference equations代考|Math462

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|偏微分方程代写partial difference equations代考|The Fourier transform

We start with the Fourier transform, which was developed by Jean Baptiste Joseph Fourier in 1822 in his work Théorie analytique de la chaleur, already mentioned earlier. Here we will take a practical approach to Fourier transforms: we will show how they can be used to solve equations explicitly. Later, in Chapter 6, they will be used for a more systematic investigation in the context of Sobolev spaces.

Unlike in the previous sections, here we consider complex-valued functions. For $1 \leq p<\infty$ we define
$$L_p(\mathbb{R}, \mathbb{C}):=\left{f: \mathbb{R} \rightarrow \mathbb{C} \text { measurable }: \int_{\mathbb{R}}|f(t)|^p d t<\infty\right} .$$
If we identify functions which coincide almost everywhere, then $L_p(\mathbb{R}, \mathbb{C})$ becomes a Banach space when equipped with the norm
$$|f|_p:=\left(\int_{\mathbb{R}}|f(t)|^p d t\right)^{1 / p} .$$
The space $L_2(\mathbb{R}, \mathbb{C})$ is a Hilbert space.
Remark 3.49 A remark about our notation is in order: the above spaces are often denoted by $L^p(\Omega, \mathbb{C})$ in the literature. We will write $p$ (the power appearing in the integrand) as a subscript to distinguish it from the order of differentiation as appears in spaces like $C^k$ and $H^k$. In dimension one we also avoid double brackets, that is, we write $L_2(0,1)$ instead of $L_2((0,1))$ (for example), even though the latter would be more consistent.

## 数学代写|偏微分方程代写partial difference equations代考|Properties of the Fourier transform

We now wish to collect a few essential properties of Fourier transforms; we will focus on those we will need when solving partial differential equations.
Theorem 3.53 (Properties and rules of calculation) Let $f \in L_1(\mathbb{R}, \mathbb{C})$. Then:
(i) The Fourier transform $\hat{f}: \mathbb{R} \rightarrow \mathbb{C}$ is continuous and $\lim {|\omega| \rightarrow \infty} \hat{f}(\omega)=0$. (ii) Linearity: if $f_1, \ldots, f_n \in L_1(\mathbb{R}, \mathbb{C})$ and $c_1, \ldots, c_n \in \mathbb{C}$, then $$\mathcal{F}\left(\sum{k=1}^n c_k f_k\right)=\sum_{k=1}^n c_k \mathcal{F} f_k .$$
(iii) If $f$ is continuously differentiable with $f^{\prime} \in L_1(\mathbb{R}, \mathbb{C})$ and $\lim {|t| \rightarrow \infty} f(t)=0$, then $$\mathcal{F}\left(f^{\prime}\right)(\omega)=i \omega \mathcal{F} f(\omega) .$$ (iv) If $\int{-\infty}^{\infty}|t f(t)| d t<\infty$, then
$$\frac{d}{d \omega} \mathcal{F} f(\omega)=(-i) \mathcal{F}(\cdot f(\cdot))(\omega) .$$
(v) For any $\alpha \in \mathbb{R}$ we have $\mathcal{F}(f(\cdot-\alpha))(\omega)=e^{-i \alpha \omega} \mathcal{F} f(\omega)$.
(vi) For any $\alpha \in \mathbb{R} \backslash{0}$ we have $\mathcal{F}(f(\alpha \cdot))(\omega)=\frac{1}{|\alpha|} \mathcal{F} f\left(\frac{\omega}{\alpha}\right)$.
Proof Here we will only prove (iii) and (iv), as these two properties will play a central role in what follows. The other statements are left to the reader as an exercise (see Exercise 3.10).
(iii) For any $R \geq 0$, if we integrate by parts and use the assumptions of the theorem, then we have
$$\frac{1}{\sqrt{2 \pi}} \int_{-R}^R f^{\prime}(t) e^{-i \omega t} d t=\left.\frac{1}{\sqrt{2 \pi}} f(t) e^{-i \omega t}\right|{t=-R} ^R+\frac{i \omega}{\sqrt{2 \pi}} \int{-R}^R f(t) e^{-i \omega t} d t .$$
By assumption, the first term converges to 0 as $R \rightarrow \infty$, while the second converges to $i \omega \hat{f}(\omega)$.

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|The Fourier transform

L_p $(\backslash m a t h b b{R}, \backslash m a t h b b{C}):=\backslash$ left ${f: \backslash m a t h b b{R} \backslash r i g h t a r r o w \backslash m a t h b b{C} \backslash$ text ${$ 可测}: \int_{ ${\operatorname{mathbb}{\mathrm{R}}}|f(t)|$

$$|f|p:=\left(\int{\mathbb{R}}|f(t)|^p d t\right)^{1 / p}$$

## 数学代写|偏微分方程代写partial difference equations代考|Properties of the Fourier transform

(i) 傅里叶变换 $\hat{f}: \mathbb{R} \rightarrow \mathbb{C}$ 是连续的并且 $\lim |\omega| \rightarrow \infty \hat{f}(\omega)=0$. (ii) 线性度: 如果
$f_1, \ldots, f_n \in L_1(\mathbb{R}, \mathbb{C})$ 和 $c_1, \ldots, c_n \in \mathbb{C}$ ，然后
$$\mathcal{F}\left(\sum k=1^n c_k f_k\right)=\sum_{k=1}^n c_k \mathcal{F} f_k$$
(iii) 如果 $f$ 连续可微 $f^{\prime} \in L_1(\mathbb{R}, \mathbb{C})$ 和 $\lim |t| \rightarrow \infty f(t)=0$ ，然后
$$\mathcal{F}\left(f^{\prime}\right)(\omega)=i \omega \mathcal{F} f(\omega)$$
(iv) 如果 $\int-\infty^{\infty}|t f(t)| d t<\infty$ ， 然后
$$\frac{d}{d \omega} \mathcal{F} f(\omega)=(-i) \mathcal{F}(\cdot f(\cdot))(\omega)$$
(v) 对于任何 $\alpha \in \mathbb{R}$ 我们有 $\mathcal{F}(f(\cdot-\alpha))(\omega)=e^{-i \alpha \omega} \mathcal{F} f(\omega)$.
(vi) 对于任何 $\alpha \in \mathbb{R} \backslash$ 我们有 $\mathcal{F}(f(\alpha \cdot))(\omega)=\frac{1}{|\alpha|} \mathcal{F} f\left(\frac{\omega}{\alpha}\right)$.

(iii) 对于任何 $R \geq 0$ ，如果我们分部积分并使用定理的假设，那么我们有
$$\frac{1}{\sqrt{2 \pi}} \int_{-R}^R f^{\prime}(t) e^{-i \omega t} d t=\frac{1}{\sqrt{2 \pi}} f(t) e^{-i \omega t} \mid t=-R^R+\frac{i \omega}{\sqrt{2 \pi}} \int-R^R f(t) e^{-i \omega t} d t$$

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