### 数学代写|复分析作业代写Complex function代考|KMA152

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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|复分析作业代写Complex function代考|One Dimensional Core-Adaptive Fourier Decomposition

Due to the above-mentioned reason we decide to use the rational orthonormal system, or by another name the Takenaka-Malmquist, or TM system in brief, introduced in Theorem 2.2. We note that TM systems in general cannot be avoided for they are Gram-Schmidt (G-S) orthogonalization of the partial fractions with poles outside the closed unit disc, the latter being fundamental constructive building blocks of rational functions in the Hardy spaces. TM systems consist of functions of positive frequency due to their construction in (finite) Blaschke products. The difference between our use and the traditional use of TM systems is that we make the parameters defining the system to be adaptive: For every individual function or signal we expand it by using a suitable TM system while the determining parameters are deliberately selected according to the data of the given function. The TM system itself may not be a basis. Whether or not the system in use is a basis, is, in fact, not interested or required. On the other hand, the adaptive expansion in the selected TM system converges very fast. And, additionally, each expanding term has positive non-constant and non-linear instantaneous frequencies. In contrast, the traditional use of a TM system is based on a fixed collection of parameters making the corresponding TM system a basis of the underlying space. The reason of use of a particular and fixed collection of parameters, however, is, as usual, not be well justified. Laguerre and two-parameter Kautz systems are examples of such fixedparameter TM bases.

In the sequel we change our function notation $s^{+}$in the Hardy $H^2(\mathbf{D})$ to $f$. In the unit circle context we have $f(z)=\sum_{l=1}^{\infty} c_l z^l, \sum_{l=1}^{\infty}\left|c_l\right|^2<\infty$. Now we seek a decomposition of $f$ into a TM system with adaptively selected parameters. The collection of the functions
$$e_a(z)=\frac{\sqrt{1-|a|^2}}{1-\bar{a} z}, \quad a \in \mathbf{D},$$
consists of normalized Szegö kernels of the disc. Below we present AFD, or more specifically, Core-AFD algorithm. Set $f=f_1$. First write
$$f(z)=\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)+\frac{f_1(z)-\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)}{\frac{z-a_1}{1-\bar{a}_1 z}} \frac{z-a_1}{1-\bar{a}_1 z}$$

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

Let $f=h g$, where $f, g$ are Hardy $H^2(\mathbf{D})$ functions, and $h$ is an inner function. Let $f$ and $g$ be expanded into their respective Fourier series, viz.,
$$f(z)=\sum_{k=0}^{\infty} c_k z^k, \quad g(z)=\sum_{k=0}^{\infty} d_k z^k$$

The Plancherel theorem and the modular 1 property of inner functions assert that
$$\sum_{k=0}^{\infty}\left|c_k\right|^2=|f|^2=|g|^2=\sum_{k=0}^{\infty}\left|d_k\right|^2$$
In digital signal processing (DSP) there is the following result: For any $n$,
$$\sum_{k=n}^{\infty}\left|c_k\right|^2 \geq \sum_{k=n}^{\infty}\left|d_k\right|^2$$
(see, for instance, $[11,19]$ ).
In DSP this is referred as energy-front-loading property of minimum phase signals. This amounts to saying that through factorizing out the inner function factor the convergence rate of the Fourier series of the remaining outer function becomes higher. This fact suggests that the AFD process would be better to incorporate with the factorization process for speeding up the convergence. This instructs that when a signal by its nature is of high frequency, one should first perform “unwinding” before extracting out from it a maximal portion of lower frequency. We proceed as follows [74, 92]. First we do factorization $f=f_1=I_1 O_1$, where $I_1$ and $O_1$ are, respectively, the inner and outer factors of $f$. The factorization is based on Nevanlinna’s factorization theorem, also see [117]. The outer function has the explicit integral representation
$$O_1(z)=e^{\frac{1}{2 \pi} \int_0^{2 \pi} \frac{e^{i t}+z}{e^{i t}-z} \log \left|f_1\left(e^{i t}\right)\right| d t}$$
The outer function is computed by using the boundary value of $f_1$. On the boundary the above integral is taken to be of the principal integral sense. The imaginary part of the integral reduces to the circular Hilbert transform of $\log \left|f_1\left(e^{i t}\right)\right|$. Next, we do a maximal sifting to $O_1$.

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|One Dimensional Core-Adaptive Fourier Decomposition

$$e_a(z)=\frac{\sqrt{1-|a|^2}}{1-\bar{a} z}, \quad a \in \mathbf{D}$$

$$f(z)=\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)+\frac{f_1(z)-\left\langle f_1, e_{a_1}\right\rangle e_{a_1}(z)}{\frac{z-a_1}{1-\bar{a}_1 z}} \frac{z-a_1}{1-\bar{a}_1 z}$$

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

$$f(z)=\sum_{k=0}^{\infty} c_k z^k, \quad g(z)=\sum_{k=0}^{\infty} d_k z^k$$
Plancherel 定理和内部函数的模 1 属性断言
$$\sum_{k=0}^{\infty}\left|c_k\right|^2=|f|^2=|g|^2=\sum_{k=0}^{\infty}\left|d_k\right|^2$$

$$\sum_{k=n}^{\infty}\left|c_k\right|^2 \geq \sum_{k=n}^{\infty}\left|d_k\right|^2$$
(例如，参见 $[11,19]$ ).

$$O_1(z)=e^{\frac{1}{2 \pi} \int_0^{2 \pi} \frac{e^{i t}+z}{e^{i t}-z} \log \left|f_1\left(e^{i t}\right)\right| d t}$$

## 有限元方法代写

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## MATLAB代写

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