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

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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|Related Studies and Applications

The AFD type expansions is in a great extent related to the Beurling-Lax shiftinvariant subspaces of the Hardy $H^2$ spaces. In the unit disc case,
$$H^2(\mathbf{D})=\overline{\operatorname{span}}\left{B_k\right}_{k=1}^{\infty} \oplus \phi H^2(\mathbf{D}),$$
where $\left{B_k\right}_{k=1}^{\infty}$ is the TM system generated by a sequence $\left{a_1, \cdots, a_n, \cdots\right}$, where multiples are counted, and $\phi$ is the Blaschke product with the zeros $\left{a_1, \cdots, a_n, \cdots\right}$ including the multiples. Note that when a Blaschke product $\phi$ having $a_k$ ‘s as all its zeros does not exist, corresponding to the condition
$$\sum_{k=1}^{\mathbf{1 2})}\left(1-\left|a_k\right|\right)<\infty,$$
then the associated TM system is a basis. Although this has been well known over a long time, its relations with adaptive expansions, as far as what are aware by the author, have not been brought up. The fact that TM systems being Schauder systems was proved in [93]. The space decomposition relation (26) was extended to $H^p$ spaces, where $p \neq 2$ [80]. Relations between backward shift invariant subspaces and bandlimited functions and Bedrosian identity $[80,107]$ were studied. There are open questions on whether there exist adaptive and fast converging expansions by using TM systems for the cases $p \neq 2$, and for $p=2$ how far one can extend AFD (26) to higher dimensions. The study has a great room to be further developed.

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

The phase and frequency studies in mono-component function theory lay certain foundations in digital signal processing. In related studies what is called extra-strong uncertainty principle
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int _ { – \infty } ^ { \infty } \left|t-\langle t\rangle\left|\left.|\phi(t)-\langle\omega\rangle || f(t)\right|^2 d t\right)^2\right.\right.$$
was recently established [22], where $f$ is a real-valued signal, $\sigma_t^2$ and $\sigma_\omega^2$ are the standard deviations with respect to the time and the Fourier frequency, and $\langle t\rangle$ and $\langle\omega\rangle$ are the corresponding means. A weaker uncertainty principle of the same type was previously given by L. Cohen
$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. We further extended the above result to multi-dimensional contexts [21-24, 26].

The Dirac-type time-frequency distribution (DTFD) of the form
$$P(t, \omega)=\rho^2(t) \delta\left(\omega-\theta^{\prime}(t)\right)$$
is the ultimate desire of signal analysts. Several time-frequency distributions, including windowed Fourier transform and Wigner-Ville transform, etc., have been used by signal analysts, of which none are entirely satisfied. The existing timefrequency distributions do not give explicit and clear frequency components, and, they often depend on parameter selections. Positive-frequency decompositions of signals offered by the AFD decompositions naturally give rise to Dirac-type timefrequency distributions. For a single mono-component $m_1(t)=\rho_1(t) \cos \theta_1(t)$ the corresponding DTFD according to (28) is the graph $\left(t, \theta_1^{\prime}(t)\right)$ of the function $\omega=\theta_1^{\prime}(t)$ in the $\omega-t$ plane, while the weight $\rho_1^2(t)$ may be represented by colors continuously changing along with changing of the values $\rho_1^2(t)$. If a signal $f$ is expanded into a series of “intrinsic composing” mono-components, then its DTFD is the bunch of color-weighted graphs of which each is made from a composing monocomponent $[20,126]$. This definition has been interested and being paid attention by signal analysts including Leon Cohen and Lorenzo Galleani, etc., and has been used in practice (see below the application section).

# 复分析代写

## 数学代写|复分析作业代写Complex function代考|Related Studies and Applications

AFD 类型展开在很大程度上与 Hardy 的 Beurling-Lax 位移不变子空间相关 $H^2$ 空间。在单元盘盒中，
$\mathrm{H}^{\wedge} 2(\backslash \mathrm{mathbf}{\mathrm{D}})=$ loverline ${\backslash 0$ peratorname ${$ span $}} \backslash l$ eft $\left{B_{-} k \backslash \text { ight }\right}_{-}{\mathrm{k}=1}^{\wedge}{\backslash$ infty $} \backslash$ plus $\backslash p h i \mathrm{H}^{\wedge} 2(\backslash \mathrm{mathbf}{\mathrm{D}})$, , 算倍数，并且 $\phi$ 是零点的 Blaschke 积 Ileft{a_1, Icdots，a_n, Icdots\right } } \text { 包括倍数。请注意，当 Blaschke } 产品 $\phi$ 有 $a_k$ 的因为它的所有雩都不存在，对应于条件
$$\sum_{k=1}^{12)}\left(1-\left|a_k\right|\right)<\infty$$

## 数学代写|复分析作业代写Complex function代考|Extra-Strong Uncertainty Principle

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left(\int_{-\infty}^{\infty}|t-\langle t\rangle||\phi(t)-\langle\omega\rangle||f(t)|^2 d t\right)^2$$

$$\sigma_t^2 \sigma_\omega^2 \geq \frac{1}{4}+\left.\left.\left|\int_{-\infty}^{\infty}(t-\langle t\rangle)(\phi(t)-\langle\omega\rangle)\right| f(t)\right|^2 d t\right|^2$$
[13]. 我们进一步将上述结果扩展到多维上下文 [21-24, 26]。

$$P(t, \omega)=\rho^2(t) \delta\left(\omega-\theta^{\prime}(t)\right)$$

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