# 数学代写|复分析作业代写Complex function代考|MATH307

## 数学代写|复分析作业代写Complex function代考|Ellipticity for 3rd-Order Higher Spin Operator D3

Theorem 8 The 3 rd-order higher spin operator, which is explicitly given by
$$\mathcal{D}_3=R_k^3-\frac{4}{(m+2 k)(m+2 k-4)} T_k T_k^* R_k,$$
is an elliptic operator if $m>4$.
Proof The technique used here is motivated by [6]. The critical point in this technique is the following: when proving the principal symbol is a linear map from $\mathcal{M}_k$ to $\mathcal{M}_k$, we choose a basis obtained from the classical CK extension for monogenic polynomials [19]. This helps us to see that the symbol is an injective map. On the other hand, the symbol is obviously linear, which completes our proof. Let $R_k(x), T_k(x)$, and $T_k^(x)$ be the symbols of $R_k, T_k$, and $T_k^$, respectively. We will show that for fixed $x \in \mathbb{R}^m$, the symbol of $\mathcal{D}_3$, which is given by
$$\sigma_x\left(\mathcal{D}_3\right)=R_k(x)^3-\frac{4}{(m+2 k)(m+2 k-4)} T_k(x) T_k^(x) R_k(x): \mathcal{M}_k \longrightarrow \mathcal{M}_k,$$ is a linear isomorphism. Since this symbol is obviously a linear map, it remains to be showed that this map is injective. Notice that $$\sigma_x\left(\mathcal{D}_3\right)=\left(R_k(x)^2-\frac{4}{(m+2 k)(m+2 k-4)} T_k(x) T_k^(x)\right) R_k(x): \mathcal{M}_k \longrightarrow \mathcal{M}_k$$
and $R_k$ is an elliptic operator [16]. Therefore, we only need to show what the term in the parenthesis above
\begin{aligned} & \sigma_x\left(\mathcal{D}_3\right)^{\prime}:=R_k(x)^2-\frac{4}{(m+2 k)(m+2 k-4)} T_k(x) T_k^*(x) \ =&-|x|^2+\frac{4 u x\left\langle x, D_u\right\rangle}{(m+2 k)(m+2 k-4)}+\frac{4(m+2 k-2)\langle u, x\rangle\left\langle x, D_u\right\rangle}{(m+2 k)(m+2 k-4)} \ &-\frac{4|u|^2\left\langle x, D_u\right\rangle^2}{(m+2 k)(m+2 k-4)}: \mathcal{M}_k \longrightarrow \mathcal{M}_k, \end{aligned}

## 数学代写|复分析作业代写Complex function代考|From Weierstrass and Runge to Mergelyan

In this and the following two sections we survey the main achievements of the classical holomorphic approximation theory. More comprehensive surveys of this subject are available in $[25,69-72,75,77,78,180]$, among other sources.

The approximation theory for holomorphic functions has its origin in two classical theorems from 1885. The first one, due to K. Weierstrass [170], concerns the approximation of continuous functions on compact intervals in $\mathbb{R}$ by polynomials.
Theorem 1 (Weierstrass (1885), [170]) Suppose $f$ is a continuous function on a closed bounded interval $[a, b] \subset \mathbb{R}$. For every $\epsilon>0$ there exists a polynomial $p$ such that for all $x \in[a, b]$ we have $|f(x)-p(x)|<\epsilon$.

Proof We use convolution with the Gaussian kernel. After extending $f$ to a continuous function on $\mathbb{R}$ with compact support, we consider the family of entire functions $$f_\epsilon(z)=\frac{1}{\epsilon \sqrt{\pi}} \int_{\mathbb{R}} f(x) e^{-(x-z)^2 / \epsilon^2} d x, \quad z \in \mathbb{C}, \epsilon>0 .$$
As $\epsilon \rightarrow 0$, we have that $f_\epsilon \rightarrow f$ uniformly on $\mathbb{R}$. Hence, the Taylor polynomials of $f_\epsilon$ approximate $f$ uniformly on compact intervals in $\mathbb{R}$. If furthermore $f$ is of class $\mathscr{C}^k$, then by a change of variable $u=x-z$ and placing the derivatives on $f$ it follows that we get convergence also in the $\mathscr{C}^k$ norm.

The paper by A. Pinkus 136 contains a more complete survey of Weierstrass’s results and of his impact on the theory of holomorphic approximation. As we shall see in Section 6.1, the idea of using convolutions with the Gaussian kernel gives major approximation results also on certain classes of real submanifolds in complex Euclidean space $\mathbb{C}^n$ and, more generally, in Stein manifolds.

One line of generalizations of Weierstrass’s theorem was discovered by M. Stone in 1937, [154, 155]. The Stone-Weierstrass theorem says that, if $X$ is a compact Hausdorff space and $A$ is a subalgebra of the Banach algebra $\mathscr{C}(X, \mathbb{R})$ which contains a nonzero constant function, then $A$ is dense in $\mathscr{C}(X, \mathbb{R})$ if and only if it separates points. It follows in particular that any complex valued continuous function on a compact set $K \subset \mathbb{C}$ can be uniformly approximated by polynomials in $z$ and $\bar{z}$. Stone’s theorem opened a major direction of research in Banach algebras.
Another line of generalizations concerns approximation of continuous functions on curves in the complex plane by holomorphic polynomials and rational functions. This led to Mergelyan and Carleman theorems discussed in the sequel.

However, we must first return to the year 1885 . The second of the two classical approximation theorems proved that year is due to C. Runge [144].

# 复分析代考

## 数学代写|复分析作业代写Complex function代考|Ellipticity for 3rd-Order Higher Spin Operator D3

$$\mathcal{D}3=R_k^3-\frac{4}{(m+2 k)(m+2 k-4)} T_k T_k^* R_k,$$ 是椭圆算子如果 $m>4$. 证明 这里使用的技术是由 [6] 激发的。该技术的关键点如下：当证明主要符号是来自 $\mathcal{M}_k$ 至 $\mathcal{M}_k$ ，我们 选择从单基因多项式的经典 CK 扩展中获得的基础 [19]。这有助于我们看出符号是单射映射。另一方 面，符号显然是线性的，这就完成了我们的证明。让 $R_k(x), T_k(x)$ ，和 $\left.T_k^{(} x\right)$ 成为的象征 $R_k, T_k$ ，和 $T{-} k^{\wedge}$ ，分别。我们将证明对于固定的 $x \in \mathbb{R}^m$ ，的符号 $\mathcal{D}_3$ ，这是由

$$\sigma_x\left(\mathcal{D}_3\right)=\left(R_k(x)^2-\frac{4}{(m+2 k)(m+2 k-4)} T_k(x) T_k^{(x)}\right) R_k(x): \mathcal{M}_k \longrightarrow \mathcal{M}_k$$

$$\sigma_x\left(\mathcal{D}_3\right)^{\prime}:=R_k(x)^2-\frac{4}{(m+2 k)(m+2 k-4)} T_k(x) T_k^*(x)=\quad-|x|^2+\frac{4 u x\left\langle x, D_u\right\rangle}{(m+2 k)(m+2 k-4)}$$

## 数学代写|复分析作业代写Complex function代考|From Weierstrass and Runge to Mergelyan

A. Pinkus 136的论文包含了对 Weierstrass 的结果及其对全纯近似理论的影响的更完整的调查。正如我们 将在第 $6.1$ 节中看到的，使用高斯核卷积的想法也给出了复欧几里得空间中某些类实数子流形的主要近 似结果 $\mathbb{C}^n$ 更一般地，在 Stein 流形中。

M. Stone 于 1937 年发现了 Weierstrass 定理的一种推广，[154，155]。Stone-Weierstrass 定理说，如果 $X$ 是紧致的膏斯多夫空间，并且 $A$ 是 Banach 代数的子代数 $\mathscr{C}(X, \mathbb{R})$ 其中包含一个非零常数函数，然后 $A$ 密集在 $\mathscr{C}(X, \mathbb{R})$ 当且仅当它分隔点。特别是紧集上的任何复值连续函数 $K \subset \mathbb{C}$ 可以由多项式统一逼近 $z$ 和 $\bar{z}$. 斯通定理开辟了巴拿赫代数的一个主要研究方向。

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