### 数学代写|偏微分方程代写partial difference equations代考|MATH1470

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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|Basics on Distributions in Euclidean Space

Let $\Omega$ be an open subset of $\mathbb{R}^n$, as before. If $u$ is a complex-valued linear functional on the vector space $C_{\mathrm{c}}^{\infty}(\Omega)$, i.e., if $u$ is a linear map $C_{\mathrm{c}}^{\infty}(\Omega) \longrightarrow \mathbb{C}$, we denote by $\langle u, \varphi\rangle$ its evaluation at the test-function $\varphi \in C_{\mathrm{c}}^{\infty}(\Omega)$. The linear functional $u$ is a distribution in $\Omega$ if $\left\langle u, \varphi_j\right\rangle \rightarrow 0$ whenever the sequence $\left{\varphi_j\right}_{j=0,1,2, \ldots} \subset C_{\mathrm{c}}^{\infty}(\Omega)$ converges to zero in the following sense:
(•) all derivatives $\partial^\alpha \varphi_j$ converge uniformly to zero and there is a compact set $K \subset \Omega$ such that $\operatorname{supp} \varphi_j \subset K$ whatever $j$.

The space of distributions in $\Omega$ is denoted by $\mathcal{D}^{\prime}(\Omega)$. The restriction of a distribution $u \in \mathcal{D}^{\prime}(\Omega)$ to an open subset $\Omega^{\prime}$ of $\Omega$ is simply the restriction of the linear functional $u$ to the linear subspace $C_{\mathrm{c}}^{\infty}\left(\Omega^{\prime}\right)$ of $C_{\mathrm{c}}^{\infty}(\Omega)$. By using partitions of unity in $C_{\mathrm{c}}^{\infty}(\Omega)$ it is readily proved that there is a smallest closed subset of $\Omega$, called the support of $u$ and denoted by supp $u$, such that $u$ vanishes (“identically”) in $\Omega \backslash F$. The subspace of distributions in $\Omega$ that have compact support (contained in $\Omega$ ) is denoted by $\mathcal{E}^{\prime}(\Omega)$; it can be identified with the dual of $C^{\infty}(\Omega)$.

The convergence of a sequence of distributions $u_j\left(j \in \mathbb{Z}{+}\right)$is to be understood in the “weak sense”: $u_j \rightarrow 0$ if $\left\langle u_j, \varphi\right\rangle \rightarrow 0$ for each $\varphi \in C{\mathrm{c}}^{\infty}(\Omega)$. For $u_j \in \mathcal{E}^{\prime}(\Omega)$ to converge to zero in $\mathcal{E}^{\prime}(\Omega)$ it is moreover required that there be a compact set $K \subset \Omega$ such that $\operatorname{supp} u_j \subset K$ for all $j$.

Every continuous linear map of $C_{\mathrm{c}}^{\infty}(\Omega)$ into itself defines, by transposition, a continuous linear map of $\mathcal{D}^{\prime}(\Omega)$ into itself. Most important among these are multiplication by smooth functions in $\Omega$ and partial derivatives. If $P\left(x, \mathrm{D}x\right)$ is a linear partial differential operator with smooth coefficients in $\Omega$ we define, for arbitrary $u \in \mathcal{D}^{\prime}(\Omega), \varphi \in C{\mathrm{c}}^{\infty}(\Omega)$,
$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$
where $P\left(x, \mathrm{D}_x\right)^{\top}$ is the transpose of $P\left(x, \mathrm{D}_x\right)$ [cf. (1.3.3)].

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

As is customary, $\mathcal{S}\left(\mathbb{R}^n\right)$ stands for the (Schwartz) space of functions $\varphi \in C^{\infty}\left(\mathbb{R}^n\right)$ rapidly decaying at infinity: given arbitrary $\alpha \in \mathbb{Z}{+}^n$ and $m \in \mathbb{Z}{+}$,
$$\sup {x \in \mathbb{R}^n}\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$ A sequence of functions $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$ converges to zero if the seminorms on the left in (2.1.1) converge to zero for all choices of $m$ and $\alpha ; \mathcal{S}\left(\mathbb{R}^n\right)$ is a Fréchet space and thus its topology can be defined by (equivalent) metrics that turn it into a complete metric space. The space $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ of tempered distributions in $\mathbb{R}^n$ is the subspace of $\mathcal{D}^{\prime}\left(\mathbb{R}^n\right)$ consisting of the distributions $u$ which can be written as finite sums of distribution derivatives $$u=\sum{|\alpha| \leq m} \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$
in which the $P_\alpha$ are polynomials and the $f_\alpha$ belong, say, to $L^1\left(\mathbb{R}^n\right)$. By transposing the dense injection $C_{\mathrm{c}}^{\infty}\left(\mathbb{R}^n\right) \hookrightarrow \mathcal{S}\left(\mathbb{R}^n\right)$ the dual of $\mathcal{S}\left(\mathbb{R}^n\right)$ is identified with $\mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$. Below we often denote by $\int u(x) \varphi(x) \mathrm{d} x$ (rather than by $\langle u, \varphi\rangle$ ) the duality bracket between $u \in \mathcal{S}^{\prime}\left(\mathbb{R}^n\right)$ and $\varphi \in \mathcal{S}\left(\mathbb{R}^n\right)$.
The Fourier transform
$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$
defines a Fréchet space isomorphism of $\mathcal{S}\left(\mathbb{R}x^n\right)$ onto $\mathcal{S}\left(\mathbb{R}{\xi}^n\right)$ whose inverse is given by
$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x .$$

# 偏微分方程代写

## 数学代写|偏微分方程代写partial difference equations代考|Basics on Distributions in Euclidean Space

(•) 所有导数 $\partial^\alpha \varphi_j$ 一致收敛于零且存在紧集 $K \subset \Omega$ 这样 $\operatorname{supp} \varphi_j \subset K$ 任何 $j$.

$$\left\langle P\left(x, \mathrm{D}_x\right) u, \varphi\right\rangle=\left\langle u, P\left(x, \mathrm{D}_x\right)^{\top} \varphi\right\rangle,$$

## 数学代写|偏微分方程代写partial difference equations代考|Tempered distributions and their Fourier transforms

$$\sup x \in \mathbb{R}^n\left(1+|x|^2\right)^{\frac{1}{2} m}\left|\partial_x^\alpha \varphi(x)\right|<+\infty .$$

$$u=\sum|\alpha| \leq m \mathrm{D}^\alpha\left(P_\alpha f_\alpha\right)$$

$$\widehat{u}(\xi)=\int_{\mathbb{R}^n} \mathrm{e}^{-i x \cdot \xi} u(x) \mathrm{d} x$$

$$u(x)=(2 \pi)^{-n} \int_{\mathbb{R}^n} \mathrm{e}^{i x \cdot \xi} \widehat{u}(\xi) \mathrm{d} x$$

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