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• Statistical Inference 统计推断
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• 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|Concluding remarks

In this paper we focused on the Hilbert space case, i.e., the case where the kernels $K_{\Theta}$ and $K_{\Theta_{P}}$ aré pósitivè définité. Récall that thé indéfinitè analoguee of Pick functions – the class $\mathcal{P}{\kappa}(\mathcal{G})$ of meromorphic $\mathcal{L}(\mathcal{G})$-valued functions such that the kernel $\mathfrak{K}(z, \omega)$ in (1.25) has $\kappa$ negative squares on $\mathbb{C}^{+}$- was introduced by Kre.̃n and Langer in [32]. They also introduced the class $\mathcal{S}{\kappa}^{0}(\mathcal{G})$ of functions $S \in \mathcal{P}{\kappa}$ such that $z S(z)$ is a Pick function (that is, the kernel $\widetilde{\mathfrak{n}}(z, \omega)$ in (1.25) is positive on $\mathbb{C}{+} .$The further generalization suggested by Derkach in [20] is the class $\mathcal{S}_{\kappa}^{k}(\mathcal{G})$ of meromorphic functions $S \in \mathcal{P}{\kappa}$ such that $z S \in \mathcal{P}{k}$, that is, such that the kernels $\mathfrak{K}(z, \omega)$ and $\widetilde{\mathfrak{K}}(z, \omega)$ in (1.25) have respectively, $\kappa$ and $k$ negative squares on $\mathbb{C}^{+}$.
Via the Potapov-Ginzburg transform, the class $\mathcal{S}{\kappa}^{k}(\widetilde{\mathcal{G}})$ is translated to the multiplicative generalized Stieltjes class $\mathcal{M S}{k}^{k}(\mathcal{G})$ with two associated reproducing kernel Pontryagin spaces with reproducing kernels (1.4) and (1.30). One can consider the indefinite analogue of Problem $1.10$ concerning two given reproducing kernel Pontryagin (rather than Hilbert) spaces. Our main results – Theorem $3.1$ and Theorem $1.4$ extends literally to this setting, with an additional observation that the observability gramians $\mathcal{G}{\Pi, A, \mu}$ and $\widetilde{\mathcal{G}}{\tilde{\Pi}, A, \mu}$ have respectively $\kappa$ and $k$ negative eigenvalues. The latter characterization might be useful for general interpolation theory in the class $\mathcal{S}_{k}^{k}(\mathcal{G})$ (although some work has been done in this direction (see, e.g., $[3,7]$ ), the area is still largely open.

We also note that there are several Stieltjes-type classes defined via two positive kernels: for example (see [33]), the class $\mathcal{R}a, b$ of Pick functions $S$ such that $(z-a) S(z)$ and $(b-z) S(z)$ are also Pick functions, or the class $\mathcal{S}[a, b]$ of Pick functions $S$ such that $\frac{z-a}{b-z} S(z)$ is also in the Pick class. The multiplicative counterparts for both of these classes are defined via the Potapov-Ginzburg transform and our Theorems $3.1$ and $1.4$ are easily translated to those settings. However, the unit disk counterpart of our results is not immediately clear. Keeping in mind the disk-setting identity (1.8) characterizing reproducing kernel Hilbert spaces with kernel of the form $\frac{J-\Theta(z) J \Theta(\omega)^{*}}{1-z \bar{\omega}}$, it would be of interest to obtain disk-analogues of the other results presented here for the half-plane setting. A suitable additive class parallel to the Stieltjes class for the disk setting is the class of functions $f$ analytic in $\mathbb{D}$ and such that both
$$f(z) \text { and } \frac{z-e^{i \tau}}{1-z e^{i \tau}} f(z), \quad(\tau \in[0,2 \pi))$$
are in the Carathéodory class (have positive semidefinite real part in $\mathbb{D}$ (Section 5 in Appendix [33]). Equivalently, $f$ is a Carathéodory class function analytic on the $\operatorname{arc}\left(e^{-i \tau}, e^{i \tau}\right)$ and with
$$\Im\left(f\left(e^{i \theta}\right)\right)=-i f\left(e^{i \theta}\right) \succeq 0, \quad|\theta|<\tau .$$
These functions play the same role in trigonometric moment problem over the arc $\left(e^{-i \tau}, e^{i \tau}\right)$ as Stieltjes functions do for the Stieltjes moment problem. Such a problem was studied in $[35,36]$ and later in $[6,19]$.

## 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|Quasi boundary triples and self-adjoint extensions

In this section we first recall the notion of quasi boundary triples and their Weyl functions in the extension theory of symmetric operators from $[3,4]$. Afterwards we provide a new sufficient criterion for self-adjointness in Theorem 2.2, which is the main abstract result in this note.

In the following let $\mathcal{H}$ be a Hilbert space with inner product $(\cdot, \cdot)_{\mathcal{H}}$. The next definition is a generalization of the concept of ordinary and generalized boundary triples; cf. $[11,13,14,15,19]$.

Definition 2.1. Let $S$ be a densely defined closed symmetric operator in $\mathcal{H}$ and let $T$ be a closable operator such that $\bar{T}=S^{}$. A triple $\left{\mathcal{G}, \Gamma_{0}, \Gamma_{1}\right}$ is a quasi boundary triple for $T \subset S^{}$ if $\left(\mathcal{G},(\cdot, \cdot){\mathcal{G}}\right)$ is a Hilbert space and the linear mappings $\Gamma{0}, \Gamma_{1}: \operatorname{dom} T \rightarrow \mathcal{G}$ satisfy the following conditions (i)-(iii).
(i) The abstract second Green identity
$$(T f, g){\mathcal{H}}-(f, T g){\mathcal{H}}=\left(\Gamma_{1} f, \Gamma_{0} g\right){\mathcal{G}}-\left(\Gamma{0} f, \Gamma_{1} g\right){\mathcal{G}}$$ holds for all $f, g \in \operatorname{dom} T$. (ii) The range of $\left(\Gamma{0}, \Gamma_{1}\right)^{\top}: \operatorname{dom} T \rightarrow \mathcal{G} \times \mathcal{G}$ is dense.
(iii) The operator $A_{0}:=T\left\lceil\operatorname{ker} \Gamma_{0}\right.$ is self-adjoint in $\mathcal{H}$.
Recall from [3, 4] that for a densely defined closed symmetric operator $S$ in $\mathcal{H}$ a quasi boundary triple $\left{\mathcal{G}, \Gamma_{0}, \Gamma_{1}\right}$ exists if and only if the deficiency indices of $S$ coincide. In this case one has dom $S=\operatorname{ker} \Gamma_{0} / 1 \operatorname{ker} \Gamma_{1}$. The notion of quasi boundary triples reduces to the well-known concept of ordinary boundary triples if $T=S^{*}$. For more details we refer the reader to $[3,4]$.

Assume now that $\left{\mathcal{G}, \Gamma_{0}, \Gamma_{1}\right}$ is a quasi boundary triple for $T \subset S^{*}$. In a similar way as for ordinary and generalized boundary triples in $[14,15]$ one associates the $\gamma$-field and the Weyl function. Their definition and some of their properties will now be recalled very briefly. Again we refer the reader to $[3,4]$ for a more detailed exposition. Observe first that the direct sum decomposition
$$\operatorname{dom} T=\operatorname{dom} A_{0} \dot{+} \operatorname{ker}(T-\lambda)=\operatorname{ker} \Gamma_{0} \dot{+} \operatorname{ker}(T-\lambda), \quad \lambda \in \rho\left(A_{0}\right), \quad \text { (2.2) }$$
implies that $\Gamma_{0} \mid \operatorname{ker}(T-\lambda)$ is invertible for $\lambda \in \rho\left(A_{0}\right)$. The $\gamma$-field $\gamma$ and the Weyl function $M$ are then defined as operator-valued functions on $\rho\left(A_{0}\right)$ by
$$\left.\lambda \mapsto \gamma(\lambda):=\left(\Gamma_{0}\right\rceil \operatorname{ker}(T-\lambda)\right)^{-1} \text { and } \lambda \mapsto M(\lambda):=\Gamma_{1} \gamma(\lambda),$$
respectively. It is clear from (2.2) that $\operatorname{dom} \gamma(\lambda)=\operatorname{dom} M(\lambda)=\operatorname{ran} \Gamma_{0}$ independent of $\lambda \in \rho\left(A_{0}\right)$. Moreover, the values $\gamma(\lambda)$ of the $\gamma$-field are densely defined and bounded operators from $\mathcal{G}$ into $\mathcal{H}$ such that $\operatorname{ran} \gamma(\lambda)=\operatorname{ker}(T-\lambda)$.

# 信号处理与线性系统代考

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