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## 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|Laplacians on the half-space

In this section we illustrate our abstract techniques from Section 2 by applying Corollary $2.3$ to an explicit boundary value problem. On the upper half-space $\mathbb{R}{+}^{d}=\left{x \in \mathbb{R}^{d}: x{d}>0\right}$ in $d \geq 2$ dimensions we consider the Laplacian with Robin boundary conditions $\tau_{N} f=\alpha \tau_{D} f$ on $\partial \mathbb{R}{+}^{d} \simeq \mathbb{R}^{d-1}$ involving an unbounded parameter function $\alpha: \mathbb{R}^{d-1} \rightarrow \mathbb{R}$. Here $\tau{D}$ and $\tau_{N}$ denote the Dirichlet and Neumann trace operator, respectively.

In order to construct a suitable quasi boundary triple consider the operators $T f=-\Delta f, \quad \operatorname{dom} T=\left{f \in H^{3 / 2}\left(\mathbb{R}{+}^{d}\right): \Delta f \in L^{2}\left(\mathbb{R}{+}^{d}\right)\right}$,
and $S f=-\Delta f, \quad \operatorname{dom} S=\left{f \in H^{2}\left(\mathbb{R}{+}^{d}\right): \tau{D} f=\tau_{N} f=0\right}$,
as well as the boundary mappings
$$\Gamma_{0} f=\tau_{N} f \text { and } \Gamma_{1} f=\tau_{D} f, \quad f \in \operatorname{dom} T .$$
The following proposition is essentially a consequence of the properties of the Dirichlet and Neumann trace operators and can be proved with standard techniques; cf. [3, Proposition 4.6]. The form of the Weyl function is a consequence of $[20,(9.65)]$

Proposition 3.1. Let T, S, $\Gamma_{0}$ and $\Gamma_{1}$ be as above. Then $\left{L^{2}\left(\mathbb{R}^{d-1}\right), \Gamma_{0}, \Gamma_{1}\right}$ is a quasi boundary triple for $T \subset S^{*}$ such that
$$\operatorname{ran} \Gamma_{0}=L^{2}\left(\mathbb{R}^{d-1}\right) \quad \text { and } \quad \operatorname{ran} \Gamma_{1}=H^{1}\left(\mathbb{R}^{d-1}\right) .$$
Furthermore, $A_{0}=T \mid \operatorname{ker} \Gamma_{0}$ coincides with the Neumann Laplacian
$$A_{N} f=-\Delta f, \quad \operatorname{dom} A_{N}=\left{f \in H^{2}\left(\mathbb{R}{+}^{d}\right): \tau{N} f=0\right},$$ and the corresponding Weyl function is given by
$$M(\lambda)=\left(-\Delta_{\mathbb{R}^{d-1}}-\lambda\right)^{-\frac{1}{2}}, \quad \lambda \in \mathbb{C} \backslash[0, \infty),$$
where $\Delta_{\mathbb{R}^{d-1}}$ is the self-udjoint Laplaciun in $L^{2}\left(\mathbb{R}^{u l-1}\right)$ with domain $H^{2}\left(\mathbb{R}^{|^{l-1}}\right)$.

## 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|Graph Laplace and Markov Operators

As is known, this setting is as follows: (a) the graph Laplacians will have positive spectrum; and (b) the transition operators (generalized Perron-Frobenius operators) will be positive, in that they map positive functions to positive functions. However the setting of these studies is discreté; as is clear for example for graphs and networks. In other words, we have countable discrete sets of vertices and edges; and so the relevant Hilbert spaces will be defined from counting measures, weighted or not.

Nonetheless, there are many important applications where the framework of countable discrete sets of vertices $V$ and edges $E$ is much too restrictive. The list of applications is long, both connections to probability, analysis, signal processing and more: graphons (limits of finite graphs), determinantal processes, machine learning, jump processes, integral operators, harmonic analysis etc. Certainly there is a rich variety of Markov processes where the natural setting for state space is a general measure space. It is our purpose here, in the measure theoretic setting, to make precise the duality between the two, transition operator and “graph” Laplacian.
Of course for general measure spaces, the word “graph” should perhaps be given a different meaning; see below. Starting with a Markov transition operator, in the measure-dynamic setting, what is the dual Laplacian; and vice versa?
In the countable discrete cases from network models, spectral theory and the tools of dynamics rely on a certain Hilbert space that measures “energy” and dissipation, but there, one refers to weighted counting measures on the respective sets $V$ and $E$. Our present paper deals with measure theoretic dynamics. We answer the following three questions: (i) What are the relevant measures for the general setting; (ii) What are the correct notions of positivity for both operators in the measure theoretic setting; and (iii) What is then the extended duality between transition operator and Laplacian?

Discrete and measurable settings. We begin here with precise definitions, and clarifications of the three problems. We first point out explicit parallels between the main objects in the theory of discrete networks and their counterparts defined in the measurable framework. More details are given in Section $2 .$

In this paper, we focus on the study of a measurable analogue of countable weighted networks, which are known also by names electrical or resistance networks (we will use them as synonyms). We recall that $(G, c)$ is called a weighted network if $G=(V, E)$ is a countable connected locally finite graph with no loops, and $c=c_{x y}$ is a symmetric function defined on pairs of connected vertices (a more detailed definition is given in Section 2). One can think of a countable network as a discrete measure space $(V, m)$ with the counting measure $m$. In general, the theory of weighted networks is built around two important operators acting on the space of functions $f: V \rightarrow \mathbb{R}$. They are the Laplace operator $\Delta$ and the Markov operator $P$.

# 信号处理与线性系统代考

## 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|Laplacians on the half-space

$$\Gamma_{0} f=\tau_{N} f \text { and } \Gamma_{1} f=\tau_{D} f, \quad f \in \operatorname{dom} T .$$

$$\operatorname{ran} \Gamma_{0}=L^{2}\left(\mathbb{R}^{d-1}\right) \quad \text { and } \quad \operatorname{ran} \Gamma_{1}=H^{1}\left(\mathbb{R}^{d-1}\right) .$$

\left 的分隔符缺失或无法识别

$$M(\lambda)=\left(-\Delta_{\mathbb{R}^{d} 1}-\lambda\right)^{-\frac{1}{2}}, \quad \lambda \in \mathbb{C} \backslash[0, \infty),$$

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