### 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|ELECENG4112

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等楖率论
• Advanced Mathematical Statistics 高等数理统计学
• (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

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Reproducing kernel Hilbert spaces with additional structure

In this paper we shall be interested in how additional properties of the positive kernel $K$ translate to additional structural properties of the reproducing kernel Hilbert space $\mathcal{H}{K}$. A specific form for the positive kernel $K$ of interest for us can be explained as follows. Given a Hilbert space $\mathcal{G}$, we define the unitary selfadjoint operator $$J=\left[\begin{array}{cc} 0 & i I{\mathcal{G}} \ -i I_{\mathcal{G}} & 0 \end{array}\right] \in \mathcal{L}(\mathcal{G} \oplus \mathcal{G})$$
To distinguish the summands in the direct sum $\widehat{\mathcal{G}}=\mathcal{G} \oplus \mathcal{G}$, we identify the first summand with the subspace $\mathcal{G}=\left{\left[\begin{array}{c}x \ 0\end{array}\right], x \in \mathcal{G}\right}$ of $\widehat{\mathcal{G}}$ and represent $\widehat{\mathcal{G}}$ as
$$\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G} .$$
We choose and fix a non-empty open subset $\Omega \subset \mathbb{C}$ which is symmetric about the real axis $\mathbb{R}$ and consider a Hilbert space $\mathcal{H}$ whose elements are $\widehat{\mathcal{G}}$-valued functions meromorphic in $\Omega$. Any reference to the value of a meromorphic function at $\alpha \in \Omega$ assumes that the function is analytic at $\alpha$.

Definition 1.2. We say that $\mathcal{H}$ is a space $\mathcal{H}(\Theta)$ if it admits a reproducing kernel $K_{\Theta}$ of the form
$$K_{\Theta}(z, \omega):=\frac{J-\Theta(z) J \Theta(\omega)^{}}{i(\bar{\omega}-z)}$$ for some function $\Theta$ meromorphic on $\Omega$, subject to $$\Theta(z) J \Theta(\bar{z})^{}=\Theta(\bar{z})^{*} J \Theta(z)=J \quad \text { for all } \quad z \in \Omega,$$ i.e., if H = HKΘ =: H(Θ) where KΘ is as in (1.4)–(1.5).

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|The Pick class and connections

Let us recall the Pick class $\mathcal{P}(\mathcal{G})$ (in the literature also known as NevanlinnaHerglotz class and sometimes also simply as $R$-class) consisting of $\mathcal{L}(\mathcal{G})$-valued functions holomorphic on the upper half-plane $\mathbb{C}{+}$with values there having positive semidefinite imaginary part, i.e., the functions $S: \mathbb{C}{+} \rightarrow \mathcal{L}(\mathcal{G})$ such that the kernel
$$\mathfrak{K}{S}(z, \omega)=\frac{S(z)-S(\omega)^{}}{z-\bar{\omega}}$$ is positive on $\mathbb{C}{+} .$In fact, if the kernel (1.18) is positive on a domain $\Omega \subset \mathbb{C}{+}$, it can be (uniquely) extended as a positive kernel to all of $\mathbb{C}{+}$due to the Pick interpolation theorem. It is convenient (and is consistent with Nevanlinna-Herglotz integral formula) furthermore to extend Pick functions to the lower half-plane by reflection: define $S(z)=S(\bar{z})^{}$ for $z \in \mathbb{C}^{-}$.

Let us note that the kernel $\mathfrak{K}{S}$ can be rewritten in a more aggregate form as \begin{aligned} \mathfrak{K}{S}(z, \bar{\omega}) &=\frac{\left[\begin{array}{ll} I & S(z) \end{array}\right]\left[\begin{array}{cc} 0 & i I_{\mathcal{G}} \ -i I_{\mathcal{G}} & 0 \end{array}\right]\left[\begin{array}{c} I \ S(\omega)^{} \end{array}\right]}{i(\bar{\omega}-z)} \ &=\frac{\left[\begin{array}{ll} I & S(z) \end{array}\right] \mathcal{J}{\mathcal{P}}\left[\begin{array}{c} I \ S(\omega)^{} \end{array}\right]}{i(\bar{\omega}-z)}, \quad \text { where } \quad \mathcal{J}{\mathcal{P}}=\left[\begin{array}{cc} 0 & i I_{\widehat{\mathcal{G}}} \ i I_{\widehat{\mathcal{G}}} & 0 \end{array}\right] . \end{aligned}
In case we replace $\mathcal{G}$ with $\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G}$, comparison of (1.19) with (1.6) suggests the close connection between the multiplicative Pick class $\mathcal{M} \mathcal{P}(\mathcal{G})$ and the Pick class over $\widehat{\mathcal{G}}$, i.e., $\mathcal{P}(\widehat{\mathcal{G}})$; the kernel $K_{\Theta}$ built from $\Theta$ appearing in (1.6) has exactly the same form as the kernel $\mathfrak{K}{S}$ built from $S$ appearing in (1.19), but with the aggregate signature matrix $\mathcal{J}{\mathcal{M} \mathcal{P}}$ for the class $\mathcal{M} \mathcal{P}(\mathcal{G})$ replaced by the aggregate signature matrix $\mathcal{J}{\mathcal{P}}$ for the class $\mathcal{P}(\widehat{\mathcal{G}})$. In fact there is a simple linear-fractional transformation $T{\mathcal{P G}}$ (called the Potapov-Ginzburg transformation (see [27]) which maps $\mathcal{P}(\widehat{\mathcal{G}})$ bijectively to $\mathcal{M P}(\mathcal{G})$ and which can be derived as follows.

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|Reproducing kernel Hilbert spaces with additional structure

$$J=\left[\begin{array}{lll} 0 & i I \mathcal{G}-i I_{\mathcal{G}} & 0 \end{array}\right] \in \mathcal{L}(\mathcal{G} \oplus \mathcal{G})$$

$$\widehat{\mathcal{G}}=\mathcal{G} \oplus J \mathcal{G} .$$

$$K_{\Theta}(z, \omega):=\frac{J-\Theta(z) J \Theta(\omega)}{i(\bar{\omega}-z)}$$

$$\Theta(z) J \Theta(\bar{z})=\Theta(\bar{z})^{*} J \Theta(z)=J \quad \text { for all } \quad z \in \Omega,$$

## 电子工程代写|信号处理与线性系统作业代写Signal Processing and Linear Systems代考|The Pick class and connections

$$\mathfrak{K} S(z, \omega)=\frac{S(z)-S(\omega)}{z-\bar{\omega}}$$

$\mathbb{C}+$ 由于 Pick 揷值定理。此外，通过反射将 Pick 函数扩展到下半平面很方便 (并且与 Nevanlinna-Herglotz 积分 公式一致)：定义 $S(z)=S(\bar{z})$ 为了 $z \in \mathbb{C}^{-}$.

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## MATLAB代写

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