微积分网课代修|函数代写Function theory代考|MAEN5060 Construction of the Bergman Kernel by means of an Orthonormal System

微积分网课代修|函数代写Function theory代考|MAEN5060 Construction of the Bergman Kernel by means of an Orthonormal System

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微积分网课代修|函数代写Function theory代考|MAEN5060 Construction of the Bergman Kernel by means of an Orthonormal System

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Construction of the Bergman Kernel by means of an Orthonormal System

Now we will endeavor to write down the Bergman kernel for the disk by means of an orthonormal basis for $A^{2}(D)$, that is, by applying Proposition 1.2.8. For a general $\Omega$, it can be rather difficult to actually write down a complete orthonormal system for $A^{2}(\Omega)$. Fortunately, the unit disk $D \subseteq \mathbb{C}$ has enough symmetry that we can actually pull this off.

It is not difficult to see that $\left{z^{j}\right}_{j=0}^{\infty}$ is an orthogonal system for $A^{2}(D)$. That is, the elements are pairwise orthogonal; but they are not normalized to have unit length. We may confirm the first of these assertions by noting that, if $j \neq k$, then
$$
\begin{aligned}
\left\langle z^{j}, z^{k}\right\rangle &=\iint_{D} z^{j} \overline{z^{k}} d A(z)=\int_{0}^{1} \int_{0}^{2 \pi} r^{j} e^{i j \theta} r^{k} e^{-i k \theta} d \theta r d r \
&=\int_{0}^{1} r^{j+k+1} d r \int_{0}^{2 \pi} e^{i(j-k) \theta} d \theta=0
\end{aligned}
$$
The system $\left{z^{j}\right}$ is complete: If $\left\langle f, z^{j}\right\rangle=0$ for every $j$, then $f$ will have a null power series expansion and hence be identically zero. It remains to normalize these monomials so that we have a complete orthonormal system.
We calculate that
$$
\begin{aligned}
\iint_{D}\left|z^{j}\right|^{2} d A(z) &=\int_{0}^{1} \int_{0}^{2 \pi} r^{2 j} d \theta r d r \
&=2 \pi \int_{0}^{1} r^{2 j+1} d r \
&=\pi \cdot \frac{1}{j+1}
\end{aligned}
$$

微积分网课代修|偏微分方程代写Partial Differential Equation代考|Construction of the Bergman Kernel by way of Differential Equations

It is actually possible to obtain the Bergman kernel of a domain in the plane from the Green’s function for that domain (see $[\mathrm{EPS}]$ ). Let us now summarize the key ideas. Chapter 8 , especially Section $8.2$, contains more detailed information about the Green’s function. Unlike the first two Bergman kernel constructions, the present one will work for any domain with $C^{2}$ boundary.
First, the fundamental solution for the Laplacian in the plane is the function
$$
\Gamma(\zeta, z)=\frac{1}{2 \pi} \log |\zeta-z|
$$
(in Proposition 8.1.4 we shall prove this assertion). This means that $\triangle_{\zeta} \Gamma(\zeta, z)=\delta_{z}$. [Observe that $\delta_{z}$ denotes the Dirac “delta mass” at $z$ and $\Delta_{\zeta}$ is the Laplacian in the $\zeta$ variable.] Here the derivatives are interpreted in the sense of distributions. In more prosaic terms, the condition is that
$$
\int \Gamma(\zeta, z) \cdot \Delta \varphi(\zeta) d \xi d \eta=\varphi(z)
$$
for any $C^{2}$ function $\varphi$ with compact support. We write, as usual, $\zeta=\xi+i \eta$.

微积分网课代修|函数代写Function theory代考|MAEN5060 Construction of the Bergman Kernel by means of an Orthonormal System

微积分网课代修|偏微分方程代写Partial Differential Equation代
考|Construction of the Bergman Kernel by means of an
Orthonormal System


现在我们将努力写下磁盘的伯格曼核,通过正交基 $A^{2}(D)$ ,即通过应用命题 1.2.8。对
于一般 $\Omega$ ,实际上写出一个完整的正交系统是相当困难的 $A^{2}(\Omega)$. 幸好单位盘 $D \subseteq \mathbb{C}$
有足够的对称性,我们实际上可以做到这一点。
正交的;但它们没有标准化为具有单位长度。我们可以通过以下方式确认这些断言中的
第一个,如果 $j \neq k$ ,然后
$$
\left\langle z^{j}, z^{k}\right\rangle=\iint_{D} z^{j} z^{k} d A(z)=\int_{0}^{1} \int_{0}^{2 \pi} r^{j} e^{i j \theta} r^{k} e^{-i k \theta} d \theta r d r \quad=\int_{0}^{1} r^{j+k+1} d r \int_{0}^{2 \pi} e^{i(j-k) \theta} d \theta=0
$$
因此为零。仍然需要对这些单项式进行归一化,以便我们拥有一个完整的正交系统。
我们计算得出
$$
\iint_{D}\left|z^{j}\right|^{2} d A(z)=\int_{0}^{1} \int_{0}^{2 \pi} r^{2 j} d \theta r d r \quad=2 \pi \int_{0}^{1} r^{2 j+1} d r=\pi \cdot \frac{1}{j+1}
$$


微积分网课代修|偏微分方程代写Partial Differential Equation代
考|Construction of the Bergman Kernel by way of Differential
Equations


实际上可以从该域的格林函数获得平面中域的 Bergman 核(参见 $[\mathrm{EPS}]$ )。现在让我
们总结一下关键思想。第8章,特别是第 $8.2$ ,包含有关格林函数的更多详细信息。与前
两个 Bergman 内核构造不同,现在的内核构造适用于任何具有 $C^{2}$ 边界。
首先,平面上拉普拉斯算子的基本解是函数
$$
\Gamma(\zeta, z)=\frac{1}{2 \pi} \log |\zeta-z|
$$
(在命题 8.1.4 中,我们将证明这个断言) 。这意味着 $\triangle_{\zeta} \Gamma(\zeta, z)=\delta_{z}$. [观察到 $\delta_{z}$ 表
示狄拉克“三角洲质量”在 $z$ 和 $\Delta_{\zeta}$ 是拉普拉斯算子 $\zeta$ 变量。] 这里的导数是在分布的意义上
解释的。用更平淡的术语来说,条件是
$$
\int \Gamma(\zeta, z) \cdot \Delta \varphi(\zeta) d \xi d \eta=\varphi(z)
$$
对于任何 $C^{2}$ 功能 $\varphi$ 与紧凑的支持。我们像往常一样写, $\zeta=\xi+i \eta$.

微积分网课代修|函数代写Function theory代考|MAEN5060 Construction of the Bergman Kernel by means of an Orthonormal System
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