数学代写|复分析作业代写Complex function代考|MX4557

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数学代写|复分析作业代写Complex function代考|On the uniqueness of $G_{n, k}$

We set
$$u_{ \pm}=p_{ \pm}^k, \quad v=\left(\frac{u_{+} u_{-}}{\rho}\right)^{1 / k}$$
and
$$D_{u_{ \pm}}=u_{ \pm} \frac{\partial}{\partial u_{ \pm}}, \quad D_\rho=\rho \frac{\partial}{\partial \rho} .$$
When $(z, t) \neq(w, s), \Delta_\lambda\left(\Phi_\lambda F_\lambda\right)=0$ with $\Phi_\lambda=A_{+}^{-\alpha_{+}} A_{-}^{-\alpha_{-}}$. This yields
\begin{aligned} {\left[D_{u_{+}} D_{u_{-}}\right.} & +v D_\rho\left(D_\rho+D_{u_{+}}+D_{u_{-}}+\beta-1\right) \ & \left.-v \rho\left(D_\rho+D_{u_{+}}+\alpha_{+}\right)\left(D_\rho+D_{u_{-}}+\alpha_{-}\right)\right] F_\lambda=0, \end{aligned}
where we defined
$$\beta=\frac{n+k-1}{k}$$
see (3.14) of [2]. Next, with a slight abuse of notation, we have
Lemma 2.5. The function
$$F\left(u_{+}, u_{-}, \rho\right)=\int_0^1 \sigma^{\alpha_{+}}(1-\sigma)^{\alpha_{-}} G\left(\sigma u_{+},(1-\sigma) u_{-}, \sigma(1-\sigma) \rho\right) \frac{d \sigma}{\sigma(1-\sigma)}$$
satisfies equation (2.3), provided that $G$ satisfies the equation
\begin{aligned} & D_{u_{+}} D_{u_{-}} G+v D_\rho\left(D_\rho+D_{u_{+}}+D_{u_{-}}+\beta-1\right) G \ & \quad-v \rho\left(2 D_\rho+D_{u_{+}}+D_{u_{-}}+\beta\right)\left(2 D_\rho+D_{u_{+}}+D_{u_{-}}+\beta+1\right) G=0 . \end{aligned}
This is Corollary 3.15 of [2]. We look for a solution of (2.7) that is a function of the variables $\rho$ and $y=p_{+}+p_{-}$only. Acting on such functions the operator that occurs in (2.7) is $v T_{n, k}$, where
\begin{aligned} T_{n, k} & =\frac{\rho^{1 / k}}{k^2}\left(\frac{\partial}{\partial y}\right)^2+D_\rho\left(D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}\right) \ & -\rho\left(2 D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}+1\right)\left(2 D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}+2\right) . \end{aligned}

数学代写|复分析作业代写Complex function代考|Separating Variables for $G_{n, k}$.

We cannot find variables which separate (1.1). On the other hand $G_{n, k}$ satisfies a separable partial differential equation. We start with (2.3) and introduce the variables $\rho_{ \pm}=u_{ \pm} v^{-k / 2}$ and $v$,
$$\rho_{+}=\left(\frac{u_{+}}{u_{-}} \rho\right)^{1 / 2}, \quad \rho_{-}=\left(\frac{u_{-}}{u_{+}} \rho\right)^{1 / 2}, \quad v=\left(\frac{u_{+} u_{-}}{\rho}\right)^{1 / 2 k} .$$
Note that $\rho=\rho_{+} \rho_{-}$. Then
$$D_{u_{ \pm}}=\frac{1}{k} D_v \pm \frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right), \quad D_\rho=-\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right),$$
and (2.3) takes the form
\begin{aligned} & {\left[\left(\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right)\right)\left(\frac{1}{k} D_v-\frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right)\right)\right.} \ & \quad-v\left(\frac{1}{k} D_v-\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right)\right)\left(\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right)+\frac{n-1}{k}\right) \ & \left.\quad-v \rho_{+} \rho_{-}\left(D_{\rho_{+}}+\alpha_{+}\right)\left(D_{\rho_{-}}+\alpha_{-}\right)\right] F_\lambda=0 . \end{aligned}

复分析代写

数学代写|复分析作业代写Complex function代考|On the uniqueness of $G_{n, k}$

$$u_{ \pm}=p_{ \pm}^k, \quad v=\left(\frac{u_{+} u_{-}}{\rho}\right)^{1 / k}$$

$$D_{u_{ \pm}}=u_{ \pm} \frac{\partial}{\partial u_{ \pm}}, \quad D_\rho=\rho \frac{\partial}{\partial \rho} .$$

\begin{aligned} {\left[D_{u_{+}} D_{u_{-}}\right.} & +v D_\rho\left(D_\rho+D_{u_{+}}+D_{u_{-}}+\beta-1\right) \ & \left.-v \rho\left(D_\rho+D_{u_{+}}+\alpha_{+}\right)\left(D_\rho+D_{u_{-}}+\alpha_{-}\right)\right] F_\lambda=0, \end{aligned}

$$\beta=\frac{n+k-1}{k}$$

$$F\left(u_{+}, u_{-}, \rho\right)=\int_0^1 \sigma^{\alpha_{+}}(1-\sigma)^{\alpha_{-}} G\left(\sigma u_{+},(1-\sigma) u_{-}, \sigma(1-\sigma) \rho\right) \frac{d \sigma}{\sigma(1-\sigma)}$$

\begin{aligned} & D_{u_{+}} D_{u_{-}} G+v D_\rho\left(D_\rho+D_{u_{+}}+D_{u_{-}}+\beta-1\right) G \ & \quad-v \rho\left(2 D_\rho+D_{u_{+}}+D_{u_{-}}+\beta\right)\left(2 D_\rho+D_{u_{+}}+D_{u_{-}}+\beta+1\right) G=0 . \end{aligned}

\begin{aligned} T_{n, k} & =\frac{\rho^{1 / k}}{k^2}\left(\frac{\partial}{\partial y}\right)^2+D_\rho\left(D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}\right) \ & -\rho\left(2 D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}+1\right)\left(2 D_\rho+\frac{1}{k} D_y+\frac{n-1}{k}+2\right) . \end{aligned}

数学代写|复分析作业代写Complex function代考|Separating Variables for $G_{n, k}$.

$$\rho_{+}=\left(\frac{u_{+}}{u_{-}} \rho\right)^{1 / 2}, \quad \rho_{-}=\left(\frac{u_{-}}{u_{+}} \rho\right)^{1 / 2}, \quad v=\left(\frac{u_{+} u_{-}}{\rho}\right)^{1 / 2 k} .$$

$$D_{u_{ \pm}}=\frac{1}{k} D_v \pm \frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right), \quad D_\rho=-\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right),$$

\begin{aligned} & {\left[\left(\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right)\right)\left(\frac{1}{k} D_v-\frac{1}{2}\left(D_{\rho_{+}}-D_{\rho_{-}}\right)\right)\right.} \ & \quad-v\left(\frac{1}{k} D_v-\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right)\right)\left(\frac{1}{k} D_v+\frac{1}{2}\left(D_{\rho_{+}}+D_{\rho_{-}}\right)+\frac{n-1}{k}\right) \ & \left.\quad-v \rho_{+} \rho_{-}\left(D_{\rho_{+}}+\alpha_{+}\right)\left(D_{\rho_{-}}+\alpha_{-}\right)\right] F_\lambda=0 . \end{aligned}

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|复分析作业代写Complex function代考|Math205B

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数学代写|复分析作业代写Complex function代考|$Q_p$ and random power series

Let $\varepsilon_n(\omega)$ be a Bernoulli sequence of random variables on a probability space. This means that each $\varepsilon_n$ takes the value +1 and -1 with probability $1 / 2$.
If $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is analytic in $\Delta$, let
$$f_\omega(z)=\sum_{n=0}^{\infty} \varepsilon_n(\omega) a_n z^n .$$
We call $f_\omega$ a random power series of $f$.
For $0 \leq p \leq 1$, the weighted Dirichlet space $\mathcal{D}p$ is the space of analytic functions $f$ in $\Delta$ satisfying $$|f|{\mathcal{D}p}^2=\sum{n=1}^{\infty} n^p\left|a_n\right|^2 \approx \iint_{\Delta}\left|f^{\prime}(z)\right|^2\left(1-|z|^2\right)^{1-p} d x d y<\infty .$$
It is not difficult to show that $f \in Q_p$ if and only if
$$\sup {a \in \Delta} \iint{\Delta}\left|f^{\prime}(z)\right|^2\left(1-\left|\varphi_a(z)\right|^2\right)^p d x d y<\infty$$

(cf. [AuXiZh, Proposition 1]). Thus $Q_p \subset \mathcal{D}_{1-p}$ for $0 \leq p \leq 1$. Note that $\mathcal{D}_0$ is the Hardy space $H^2$. W. Sledd and D. Stegenga [SlSt] have proved

Theorem 6.1. There exists $f(z)=\sum_{n=0}^{\infty} a_n z^n \in H^2$ (i.e. $\left.\sum_{n=0}^{\infty}\left|a_n\right|^2<\infty\right)$ but its randomization $f_\omega \notin B M O A$ for any choice of $\omega$.

By (9) we have the necessary condition: if $f(z)=\sum_{n=0}^{\infty} a_n z^n \in Q_p$ for $0<p \leq 1$, then $\sum_{n=1}^{\infty} n^{1-p}\left|a_n\right|^2<\infty$. Surprisingly, it is also a sufficient condition, almost surely, provided $0<p<1$, as the next theorem shows. At the same time this theorem shows a different behaviour of $Q_p(0<p<1)$ as compared with $B M O A$.

Theorem 6.2. [AuStZh, Theorem 1] For any $f(z)=\sum_{n=0}^{\infty} a_n z^n \in Q_p$ (i.e. $\left.\sum_{n=1}^{\infty} n^{1-p}\left|a_n\right|^2<\infty\right), 0<p<1$, we have $f_\omega \in Q_p$ a.s. (almost surely).
Proof. Since $\sum_{n=1}^{\infty} n^{1-p}\left|a_n\right|^2<\infty, f(z)=\sum_{n=0}^{\infty} a_n z^n \in \mathcal{D}{1-p}$. By [CoShUl, Theorem 2] $f\omega \in M\left(\mathcal{D}{1-p}\right)$ a.s., where $M\left(\mathcal{D}{1-p}\right)$ is the space of pointwise multipliers of $\mathcal{D}{1-p}$. By Theorem 5.2 in [AuLaXiZh], $M\left(\mathcal{D}{1-p}\right) \subset Q_p$. Thus $f_\omega \in Q_p$ a.s., and the proof is completed.

数学代写|复分析作业代写Complex function代考|New trends in $Q_p$ research

For functions in $B M O A$ we know the following boundary criterion: If $f \in H^1$ (Hardy space), then $f \in B M O A$ if and only if
$$\sup _{I \subset \partial \Delta} \frac{1}{|I|^2} \int_I \int_I\left|f\left(e^{i \theta}\right)-f\left(e^{i t}\right)\right|^2 d \theta d t<\infty$$
$(|I|$ is the length of an arc $I \subset \partial \Delta)$.
M. Essen and J. Xiao have proved the following boundary value criterion for $Q_p$ functions:

Theorem 7.1. [EsXi] Let $0<p<1$ and let $f \in H^1$. Then $f \in Q_p$ if and only if
$$\sup _{I \subset \partial \Delta} \frac{1}{|I|^p} \int_I \int_I \frac{\left|f\left(e^{i \theta}\right)-f\left(e^{i t}\right)\right|^2}{\left|e^{i \theta}-e^{i t}\right|^{2-p}} d \theta d t<\infty .$$
This boundary value criterion might have some applications in more general settings. Further, M. Essen and J. Xiao have considered the question when the Blaschke product
$$B\left(z, z_n\right)=\prod_n \frac{\left|z_n\right|}{z_n} \frac{z_n-z}{1-\bar{z}_n z}$$
belongs to $Q_p$ for $0<p<1$. Note that not all bounded functions belong to $Q_p$, $0<p<1$. O. Resendiz and L. M. Tovar have continued this research by searching for explicit conditions on the zeros $\left{z_n\right}$ of the Blaschke product which guarantee it to belong to $Q_p$.

Since the definition (1) of the spaces $Q_p$ is Möbius invariant we can transfer it to arbitrary Riemann surfaces $R$ with Green’s functions. The nesting property $Q_p(R) \subset Q_q(R), 0<p<q<\infty$, and the inclusion $\mathcal{D}_1(R) \subset Q_p(R)$ for all $p$, $0<p<\infty$, have been proved. The latter inclusion sharpens T. Metzger’s wellknown result that the classical Dirichlet space $\mathcal{D}_1(R) \subset B M O A(R)$. Also for any $p, 1<p<\infty$, there exists a Riemann surface $R$ such that $Q_p(R) \subsetneq \mathcal{B}(R)$, where $\mathcal{B}(R)$ is the Bloch space on $R$. This differs from the situation in the unit disk $\Delta$.

复分析代写

数学代写|复分析作业代写Complex function代考|$Q_p$ and random power series

$$f_\omega(z)=\sum_{n=0}^{\infty} \varepsilon_n(\omega) a_n z^n .$$

$$\sup {a \in \Delta} \iint{\Delta}\left|f^{\prime}(z)\right|^2\left(1-\left|\varphi_a(z)\right|^2\right)^p d x d y<\infty$$

(参见[auxi，提案1])。因此，$Q_p \subset \mathcal{D}_{1-p}$代表$0 \leq p \leq 1$。注意$\mathcal{D}_0$是Hardy空间$H^2$。W. Sledd和D. Stegenga [SlSt]已经证明了

数学代写|复分析作业代写Complex function代考|New trends in $Q_p$ research

$$\sup _{I \subset \partial \Delta} \frac{1}{|I|^2} \int_I \int_I\left|f\left(e^{i \theta}\right)-f\left(e^{i t}\right)\right|^2 d \theta d t<\infty$$
$(|I|$是弧的长度$I \subset \partial \Delta)$。
M. Essen和J. Xiao证明了$Q_p$函数的边值判据如下:

$$\sup _{I \subset \partial \Delta} \frac{1}{|I|^p} \int_I \int_I \frac{\left|f\left(e^{i \theta}\right)-f\left(e^{i t}\right)\right|^2}{\left|e^{i \theta}-e^{i t}\right|^{2-p}} d \theta d t<\infty .$$

$$B\left(z, z_n\right)=\prod_n \frac{\left|z_n\right|}{z_n} \frac{z_n-z}{1-\bar{z}_n z}$$
$0<p<1$属于$Q_p$。注意，并非所有有界函数都属于$Q_p$, $0<p<1$。O. Resendiz和L. M. Tovar通过寻找Blaschke积的零$\left{z_n\right}$上保证其属于$Q_p$的明确条件，继续了这项研究。

有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

数学代写|复分析作业代写Complex function代考|MATH2242

statistics-lab™ 为您的留学生涯保驾护航 在代写复分析Complex function方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写复分析Complex function代写方面经验极为丰富，各种代写复分析Complex function相关的作业也就用不着说。

数学代写|复分析作业代写Complex function代考|Strict inclusions

Because $Q_p=Q_q=\mathcal{B}$ and $Q_{p, 0}=Q_{q, 0}=\mathcal{B}0$ for $1{p, 0}$. We will use the following lemmas in proving the main theorem of this section.

Lemma 3.1. Let $01$ for all $k$, then there is a constant $A$ depending only on $p$ and $\lambda$ such that
$$A^{-1}\left(\sum_{k=1}^{\infty}\left|a_k\right|^2\right)^{\frac{1}{2}} \leq\left(\frac{1}{2 \pi} \int_0^{2 \pi}\left|\sum_{k=1}^{\infty} a_k e^{i n_k \theta}\right|^p d \theta\right)^{\frac{1}{p}} \leq A\left(\sum_{k=1}^{\infty}\left|a_k\right|^2\right)^{\frac{1}{2}}$$
for any number $a_k, k=1,2, \ldots$.
The above lemma is due to $[\mathrm{Zy}]$.
Lemma 3.2. Let $\alpha>0, p>0, n \geq 0, a_n \geq 0, I_n=\left{k: 2^n \leq k<2^{n+1}, k \in \mathbb{N}\right}$, $t_n=\sum_{k \in I_n} a_k$ and $f(x)=\sum_{n=1}^{\infty} a_n x^n$. Then there is a constant $K$ depending only on $p$ and $\alpha$ such that
$$\frac{1}{K} \sum_{n=0}^{\infty} 2^{-n \alpha} t_n^p \leq \int_0^1(1-x)^{\alpha-1} f(x)^p d x \leq K \sum_{n=0}^{\infty} 2^{-n \alpha} t_n^p .$$
The proof of Lemma 3.2 can be found in [MaPa]. By simple calculation we see that the above lemma is still valid for $f(x)=\sum_{n=1}^{\infty} a_n x^{n-1}, a_n \geq 0$ (cf. [Mi, p. 108].
The next lemma can be found in [AuXiZh, Theorem 5].

Lemma 3.3. Let $I_n=\left{k: 2^n \leq k<2^{n+1}, k \in \mathbb{N}\right}$ and let $f(z)=\sum_{n=0}^{\infty} a_n z^n$ be analytic in $\Delta$. If, for $0<p \leq 1$,
$$\sum_{n=0}^{\infty} 2^{n(1-p)}\left(\sum_{k \in I_n}\left|a_k\right|\right)^2<\infty,$$
then $f \in Q_{p, 0}$.
The idea of the proof of the following theorem is found in [Mi, Theorem 2].

数学代写|复分析作业代写Complex function代考|$Q_p$ and some well-known function spaces

We take one example showing the relation between $Q_p$ and a well-known function space. For $1<q<\infty$, we say an analytic function $f$ in $\Delta$ is in the (analytic) Besov space $B_q$ provided
$$\iint_{\Delta}\left|f^{\prime}(z)\right|^q\left(1-|z|^2\right)^{q-2} d x d y<\infty .$$
Note that $B_2=\mathcal{D}1$ (the classical Dirichlet space). It is well known that $B_q$ increases with $q$ and $B_q \subset V M O A$ for all $1{0<p<1} Q_{p, 0}$ for $1<q \leq 2$,
(ii) $B_q \subset \bigcap_{1-\frac{2}{q}<p<1} Q_{p, 0}$ for $2 \leq q<\infty$.
It can be shown that both inclusions in Theorem 4.1 (i) and (ii) are strict and that the lower bound, $1-2 / q$, of values of $p$ for $2<q<\infty$, is best possible in the sense that
$$B_q \not \subset Q_{1-\frac{2}{q}, 0}$$
for $2<q<\infty$. As an example function we can choose $f(z)=\sum_{n=0}^{\infty} \frac{1}{(n+1)^{\frac{1}{2}} 2^{\frac{n}{q}}} z^{2^n}$.
Other inclusions between $Q_p$ and Lipschitz spaces or mean Lipschitz spaces are also known.

In Section 3 we have a criterion for a Hadamard gap series to belong to $Q_p$ for $0<p \leq 1$. If an analytic function $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is not a Hadamard gap series, is it possible to check by its Taylor coefficients that it belongs to $Q_p$ ? The next theorem gives a sufficient condition for arbitrary $a_n$ ‘s and a criterion for positive $a_n$ ‘s.

Theorem 5.1. [AuStXi, Theorem 1.2] Suppose that $0<p \leq 1$ and that $f(z)=$ $\sum_{n=0}^{\infty} a_n z^n$ is an analytic function.
(i) The condition
$$\sup k \frac{1}{k^p} \sum{n=0}^{\infty}(n+1)^{1-p}\left[\sum_{m=0}^{\min (n, k)} \frac{\left|a_{2 n-m+1}\right|}{(m+1)^{1-p}}\right]^2<\infty$$
implies that $f \in Q_p$.
(ii) If $a_n \geq 0$ for all $n$ and $f \in Q_p$ then condition (8) holds.
For $B M O A$, the corresponding criterion to (ii) is C. Fefferman’s unpublished result, whose published proofs involve some aspect of the duality between $H^1$ and $B M O A$. In the absence of an analogue to these theories, the proof of Theorem 5.1 only uses the definition of $Q_p$.

The above theorem is a powerful tool when constructing functions $f$ satisfying $f \in Q_q \backslash Q_p, 0<p<q \leq 1$, and some extra condition which excludes the use of lacunary series with Hadamard gaps.

复分析代写

数学代写|复分析作业代写Complex function代考|Strict inclusions

$$A^{-1}\left(\sum_{k=1}^{\infty}\left|a_k\right|^2\right)^{\frac{1}{2}} \leq\left(\frac{1}{2 \pi} \int_0^{2 \pi}\left|\sum_{k=1}^{\infty} a_k e^{i n_k \theta}\right|^p d \theta\right)^{\frac{1}{p}} \leq A\left(\sum_{k=1}^{\infty}\left|a_k\right|^2\right)^{\frac{1}{2}}$$

(i)条件
$$\sup k \frac{1}{k^p} \sum{n=0}^{\infty}(n+1)^{1-p}\left[\sum_{m=0}^{\min (n, k)} \frac{\left|a_{2 n-m+1}\right|}{(m+1)^{1-p}}\right]^2<\infty$$

(ii)如果$n$和$f \in Q_p$均为$a_n \geq 0$，则条件(8)成立。

有限元方法代写

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MATLAB代写

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