### 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|EE483

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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

## 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 focus here

However our focus here is not on interpolation aspects but rather on the intrinsic structure of the associated reproducing kernel Hilbert spaces. The main objective of the present paper is to find Stieltjes-class counterparts of Theorems $1.3$ and 1.4. Specifically, in Section 3 we shall consider the following:

Problem 1.10. Given two reproducing kernel Hilbert spaces $\mathcal{H}$ and $\tilde{\mathcal{H}}$ of $\widehat{\mathcal{G}}$-valued functions meromorphic in $\Omega$, find necessary and sufficient conditions for the existence of a function $\Theta \in \mathcal{M S}(\mathcal{G}, \Omega)$ such that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\widetilde{\mathcal{H}}=\mathcal{H}\left(P \Theta P^{-1}\right)$. In case $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ are presented as ranges of observability operators
$$\mathcal{H}=\operatorname{Ran} \mathcal{O}{\Pi, A, \mu} \quad \text { and } \quad \tilde{\mathcal{H}}=\operatorname{Ran} \mathcal{O}{\tilde{\Pi}, \tilde{A}, \mu},$$ find necessary and sufficient conditions directly in terms of the operators $\Pi, A, \widetilde{\Pi}, \widetilde{A}$ for it to happen that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\tilde{\mathcal{H}}=\mathcal{H}\left(\Theta_{P}\right)$ for some $\Theta$.

Solutions to these problems are presented in Theorem $3.1$ (the Stieltjes analogue of Theorem 1.3) and Theorem $4.1$ (the Stieltjes analogue of Theorem 1.4).
Finally we note that the reproducing kernel space $\mathcal{H}(\Theta)$ determines the function $\Theta \in \mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ only up to a unitary constant right factor $\Upsilon$. While $\Theta \Upsilon$ is in the Pick class $\mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ whenever $\Theta \in \mathcal{M} \mathcal{P}(\mathcal{G}, \Omega)$ for any constant $J$-unitary operator $\Upsilon$, the corresponding property for the multiplicative Stieltjes class fails in general. Thus it is a subtle but nontrivial point to show that, if $\Theta$ is such that $\mathcal{H}=\mathcal{H}(\Theta)$ and $\widetilde{\mathcal{H}}=\mathcal{H}\left(\Theta_{P}\right)$, then there is a choice of constant J-unitary operators $\Upsilon$ and $\tilde{\Upsilon}$ so that $(\Theta \cdot \Upsilon){P}=\Theta{P} \cdot \widetilde{\Upsilon}$, in which case we then have $\Theta^{\prime}:=\Theta \cdot \Upsilon \in \mathcal{M} \mathcal{S}(\mathcal{G}, \Omega)$ as well as $\mathcal{H}=\mathcal{H}\left(\Theta^{\prime}\right)$ and $\tilde{\mathcal{H}}=\mathcal{H}\left(\left(\Theta^{\prime}\right)_{P}\right)$. This issue is addressed in Section $4.2$ below.

The paper is organized as follows. Section 2 presents some material on the simultaneous $J$-unitary equivalence of a pair of Krein-space operators as well as some identities involving the operators $R_{\alpha}$ and $\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right]$ needed in the proof of the characterization of a pair of reproducing kernel Hilbert spaces of the form $\mathcal{H}(\Theta)$ and $\mathcal{H}\left(\Theta_{P}\right)$. Section 3 gives an intrinsic structural characterization of pairs of reproducing kernel Hilbert spaces of the form $\left(\mathcal{H}(\Theta), \mathcal{H}\left(\Theta_{P}\right)\right)$ in intrinsic geometric, structural form, while in Section 4, these results are reformulated in explicit state-space coordinates.

## 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|Characterization of Stieltjes reproducing-kernel

In this section we characterize pairs $\left{\mathcal{H}(\Theta), \mathcal{H}\left(P \Theta P^{-1}\right)\right}$ in terms of invariance properties and structure identities.

Theorem 3.1. Let $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ be two reproducing kernel Hilbert spaces whose elements are $\widehat{\mathcal{G}}$ valued functions which are meromorphic in $\Omega$. In order that $\mathcal{H}$ and $\widetilde{\mathcal{H}}$ be spaces $\mathcal{H}(\Theta)$ and $\mathcal{H}\left(P \Theta P^{-1}\right)$ it is necessary and sufficient that

1. For each $\alpha \in \Omega$, the invariance conditions
$$R_{\alpha} \mathcal{H} \subset \mathcal{H}, \quad R_{\alpha} \tilde{\mathcal{H}} \subset \widetilde{\mathcal{H}}$$
hold as well as the coupled invariance conditions
$$\left[\begin{array}{cc} I+\alpha R_{\alpha} & 0 \ 0 & R_{\alpha} \end{array}\right] \mathcal{H} \subset \tilde{\mathcal{H}} \text { and }\left[\begin{array}{cc} R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha} \end{array}\right] \tilde{\mathcal{H}} \subset \mathcal{H} .$$
2. The following four identities hold for all functions
3. $F=\left[\begin{array}{l}F_{1} \ F_{2}\end{array}\right] \in \mathcal{H}, \quad G=\left[\begin{array}{l}G_{1} \ G_{2}\end{array}\right] \in \mathcal{H}, \quad \widetilde{F}=\left[\begin{array}{c}\widetilde{F}{1} \ \widetilde{F}{2}\end{array}\right] \in \widetilde{\mathcal{H}}, \quad \widetilde{G}=\left[\begin{array}{c}\widetilde{G}{1} \ \widetilde{G}{2}\end{array}\right] \in \widetilde{\mathcal{H}}$
4. and for all $\alpha, \beta \in \Omega$ :
5. $\left\langle R_{\alpha} F,\left(I+\beta R_{\beta}\right) G\right\rangle_{\mathcal{H}}-\left\langle\left[\begin{array}{cc}I+\alpha R_{\alpha} & 0 \ 0 & R_{\alpha}\end{array}\right] F,\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
6. $=G_{2}(\beta)^{} F_{1}(\alpha)$, $\left\langle\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left[\begin{array}{cc}R_{\beta} & 0 \ 0 & I+\beta R_{\beta}\end{array}\right] \widetilde{G}\right\rangle_{\mathcal{H}}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F}, R_{\beta} \widetilde{G}\right\rangle_{\tilde{\mathcal{H}}}$ $=\widetilde{G}{2}(\beta)^{} \widetilde{F}{1}(\alpha)$,
7. $\left\langle\left[\begin{array}{cc}R_{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F}, R_{\beta} G\right\rangle_{\mathcal{H}}-\left\langle R_{\alpha} \widetilde{F},\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
8. $=G_{1}(\beta)^{} \widetilde{F}{2}(\alpha)$, $\left\langle\left[\begin{array}{cc}R{\alpha} & 0 \ 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left(I+\beta R_{\beta}\right) G\right\rangle_{\varkappa}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F},\left[\begin{array}{cc}I+\beta R_{\beta} & 0 \ 0 & R_{\beta}\end{array}\right] G\right\rangle_{\tilde{\varkappa}}$
9. $=G_{2}(\beta)^{} \widetilde{F}_{1}(\alpha)$

## 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|Characterization of Stieltjes reproducing-kernel

1. 对于每个 $\alpha \in \Omega$, 不变条件
$$R_{\alpha} \mathcal{H} \subset \mathcal{H}, \quad R_{\alpha} \tilde{\mathcal{H}} \subset \widetilde{\mathcal{H}}$$
保持以及耦合不变条件
$$\left[\begin{array}{llll} I+\alpha R_{\alpha} & 0 & 0 & R_{\alpha} \end{array}\right] \mathcal{H} \subset \tilde{\mathcal{H}} \text { and }\left[\begin{array}{llll} R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha} \end{array}\right] \tilde{\mathcal{H}} \subset \mathcal{H} .$$
2. 以下四个恒等式适用于所有功能
3. $F=\left[\begin{array}{ll}F_{1} & F_{2}\end{array}\right] \in \mathcal{H}, \quad G=\left[\begin{array}{ll}G_{1} & G_{2}\end{array}\right] \in \mathcal{H}, \quad \widetilde{F}=[\widetilde{F} 1 \widetilde{F} 2] \in \widetilde{\mathcal{H}}, \quad \widetilde{G}=[\widetilde{G} 1 \widetilde{G} 2] \in \widetilde{\mathcal{H}}$
4. 并为所有人 $\alpha, \beta \in \Omega$ :
5. $\left\langle R_{\alpha} F,\left(I+\beta R_{\beta}\right) G\right\rangle_{\mathcal{H}}-\left\langle\left[I+\alpha R_{\alpha} \quad 0 \quad 0 \quad R_{\alpha}\right] F,\left[I+\beta R_{\beta} \quad 0 \quad 0 \quad R_{\beta}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
6. $=G_{2}(\beta) F_{1}(\alpha)$,
$\left\langle\left[\begin{array}{llll}R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F},\left[\begin{array}{llll}R_{\beta} & 00 & I+\beta R_{\beta}\end{array}\right] \widetilde{G}\right\rangle_{\mathcal{H}}-\left\langle\left(I+\alpha R_{\alpha}\right) \widetilde{F}, R_{\beta} \widetilde{G}\right\rangle_{\tilde{\mathcal{H}}}$ $=\widetilde{G} 2(\beta) \widetilde{F} 1(\alpha)$,
7. $\left\langle\left[\begin{array}{llll}R_{\alpha} & 0 & 0 & I+\alpha R_{\alpha}\end{array}\right] \widetilde{F}, R_{\beta} G\right\rangle_{\mathcal{H}}-\left\langle R_{\alpha} \widetilde{F},\left[I+\beta R_{\beta} \quad 00 \quad R_{\beta}\right] G\right\rangle_{\tilde{\mathcal{H}}}$
8. $=G_{1}(\beta) \widetilde{F} 2(\alpha)$, 9. $=G_{2}(\beta) \widetilde{F}_{1}(\alpha)$

## 广义线性模型代考

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