# 数学代写|复分析作业代写Complex function代考|MAST30021

## 数学代写|复分析作业代写Complex function代考|Approximation on Unbounded sets in Riemann Surfaces

It seems that the first result concerning the approximation of functions on unbounded closed subsets of $\mathbb{C}$ by entire functions is the following generalization of Weierstrass’s Theorem 1, due to T. Carleman [31].

Theorem 8 (Carleman (1927), [31]) Given continuous functions $f: \mathbb{R} \rightarrow \mathbb{C}$ and $\epsilon: \mathbb{R} \rightarrow(0,+\infty)$, there exists an entire function $F \in \mathscr{O}(\mathbb{C})$ such that
$$|F(x)-f(x)|<\epsilon(x) \text { for all } x \in \mathbb{R} \text {. }$$ This says that continuous functions on $\mathbb{R}$ can be approximated in the fine $\mathscr{C}^0$ topology by restriction to $\mathbb{R}$ of entire functions on $\mathbb{C}$. The proof amounts to inductively applying Mergelyan’s theorem on polynomial approximation (Theorem 3). Proof Recall that $\overline{\mathbb{D}}={z \in \mathbb{C}:|z| \leq 1}$. For $j \in \mathbb{Z}{+}={0,1, \ldots}$ set $$K_j=j \overline{\mathbb{D}} \cup[-j-2, j+2], \quad \epsilon_j=\min {\epsilon(x):|x| \leq j+2} .$$ Note that $\epsilon_j \geq \epsilon{j+1}>0$ for all $j \in \mathbb{Z}{+}$. We construct a sequence of continuous functions $f_j:(j+1 / 3) \mathbb{D} \cup \mathbb{R} \rightarrow \mathbb{C}$ satisfying the following conditions for all $j \in \mathbb{N}$ : $\left(\mathrm{a}_j\right) f_j$ is holomorphic on $(j+1 / 3) \mathbb{D}$, $\left(\mathrm{b}_j\right) f_j(x)=f(x)$ for $x \in \mathbb{R}$ with $|x| \geq j+2 / 3$, and $\left(\mathrm{c}_j\right)\left|f_j-f{j-1}\right|<2^{-j-1} \epsilon_{j-1}$ on $K_{j-1}$.
To construct $f_0$, we pick a smooth function $\chi: \mathbb{R} \rightarrow[0,1]$ such that $\chi(x)=1$ for $|x| \leq 1 / 3$ and $\chi(x)=0$ for $|x| \geq 2$ 2/3. Mergelyan’s theorem (see Theorem 3) gives a holomorphic polynomial $h$ such that, if we define $f_0$ to equal $h$ on $(1 / 3) \mathbb{D}$ and set $f_0(x)=\chi h(x)+(1-\chi) f(x)$ for $|x| \geq 1 / 3$, then $f_0$ satisfies conditions $\left(a_0\right)$ and $\left(b_0\right)$, while condition $\left(c_0\right)$ is vacuous.

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem for C r Functions

In applications, one is often faced with the approximation problem for functions of class $\mathscr{C}^r(r \in \mathbb{N})$ on compact or closed sets in a Riemann surface. Such problems arise not only in complex analysis (for instance, in constructions of closed complex curves in complex manifolds, see [44], or in constructions of proper holomorphic embeddings of open Riemann surfaces into $\mathbb{C}^2$, see [67, 68] and [62, Chap. 9]), but also in related areas such as the theory of minimal surfaces in Euclidean spaces $\mathbb{R}^n$ (see the recent survey [3]), the theory of holomorphic Legendrian curves in complex contact manifolds (see $[2,4]$ ), and others. In most geometric constructions it suffices to consider compact sets of the following type.

Definition 3 (Admissible Sets in Riemann Surfaces) A compact set $S$ in a Riemann surface $X$ is admissible if it is of the form $S=K \cup M$, where $K$ is a finite union of pairwise disjoint compact domains with piecewise $\mathscr{C}^1$ boundaries in $X$ and $M=S \backslash \stackrel{\circ}{K}$ is a union of finitely many pairwise disjoint smooth Jordan arcs and closed Jordan curves meeting $K$ only in their endpoints (or not at all) and such that their intersections with the boundary $b K$ of $K$ are transverse.

Clearly, the complement $X \backslash S$ of an admissible set has at most finitely many connected components, and hence Theorem 5 applies.

A function $f: S=K \cup M \rightarrow \mathbb{C}$ on an admissible set is said to be of class $\mathscr{C}^r(S)$ if $\left.f\right|K \in \mathscr{C}^r(K)$ (this means that it is of class $\mathscr{C}^r(K)$ and all its partial derivatives of order $\leq r$ extend continuously to $K$ ) and $\left.f\right|_M \in \mathscr{C}^r(M)$. Whitney’s jet-extension theorem (see Theorem 46) shows that any $f \in \mathscr{A}^r(S)$ extends to a function $f \in \mathscr{C}^r(X)$ which is $\bar{\partial}$-flat to order $r$ on $S$, meaning that $$\lim {x \rightarrow S} D^{r-1}(\bar{\partial} f)(x)=0 .$$
Here, $D^k$ denotes the total derivative of order $k$ (the collection of all partial derivatives of order $\leq k$ ). We define the $\mathscr{C}^r(S)$ norm of $f$ as the maximum of derivatives of $f$ up to order $r$ at points $z \in S$, where for points $z \in M \backslash K$ we consider only the tangential derivatives. (This equals the $r$-jet norm on $S$ of a $\bar{\partial}$-flat extension of $f$.)

We have the following approximation result for functions of class $\mathscr{A}^r$ on admissible sets in Riemann surfaces. Corollary 9 in Section $7.2$ gives an analogous result for manifold-valued maps.

# 复分析代考

## 数学代写|复分析作业代写Complex function代考|Approximation on Unbounded sets in Riemann Surfaces

$$|F(x)-f(x)|<\epsilon(x) \text { for all } x \in \mathbb{R} \text {. }$$ 这表示连续函数在垚可以近似于罚款 $\mathscr{C}^0$ 限制拓扑 $\mathbb{R}$ 上的全部功能 $\mathbb{C}$. 证明相当于在多项式逼近上归纳应用 Mergelyan 定理 (定理 3) 。证明 回想一下 $\overline{\mathbb{D}}=z \in \mathbb{C}:|z| \leq 1$. 为了 $j \in \mathbb{Z}+=0,1, \ldots$ 放 $$K_j=j \overline{\mathbb{D}} \cup[-j-2, j+2], \quad \epsilon_j=\min \epsilon(x):|x| \leq j+2 .$$ 注意 $\epsilon_j \geq \epsilon j+1>0$ 对所有人 $j \in \mathbb{Z}+$. 我们构建了一系列连续函数 $f_j:(j+1 / 3) \mathbb{D} \cup \mathbb{R} \rightarrow \mathbb{C}$ 全部满 足以下条件 $j \in \mathbb{N}:\left(\mathrm{a}j\right) f_j$ 是全纯的 $(j+1 / 3) \mathbb{D} ，\left(\mathrm{~b}_j\right) f_j(x)=f(x)$ 为了 $x \in \mathbb{R}$ 和 $|x| \geq j+2 / 3$ ，和 $\left(c_j\right)\left|f_j-f j-1\right|<2^{-j-1} \epsilon{j-1}$ 上 $K_{j-1}$.

## 数学代写|复分析作业代写Complex function代考|Mergelyan’s Theorem for C r Functions

$$\lim x \rightarrow S D^{r-1}(\bar{\partial} f)(x)=0 .$$

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