# 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|ELEN30012

## 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|Structure of the energy space

We address now the question about the structure of the energy space $\mathcal{H}$. We first show that $\mathcal{H}$ contains the linear subspace spanned by characteristic functions of sets of finite measure. In the next statement we also give two formulas for computation of the norm of $\chi_{A}$ and the inner product of characteristic functions in terms of the measure $\rho$.
Lemma 6.18. Suppose $c(x)$ is locally integrable with respect to $\mu$. Then
$$\mathcal{D}{\text {fin }} \subset \mathcal{H} .$$ Moreover, if $A \in \mathcal{B}{\text {fin }}$, then
$$\left|\chi_{A}\right|_{\mathcal{H}{E}}^{2} \leq \int{A} c(x) d \mu(x),$$
and
$$\left|\chi_{A}\right|_{\mathcal{H}{E}}^{2}=\rho\left(A \times A^{c}\right)$$ where $A^{c}:=V \backslash A$. More generally, one has \begin{aligned} \left\langle\chi{A}, \chi_{B}\right\rangle_{\mathcal{H}{E}} &=\rho((A \cap B) \times V)-\rho(A \times B) \ &=\nu(A \cap B)-\rho(A \times B) \end{aligned} and $$\chi{A} \perp \chi_{B} \Longleftrightarrow \rho((A \backslash B) \times B)=\rho\left((A \cap B) \times B^{c}\right) .$$
Proof. We use (6.5) and compute for $A \in \mathcal{B}{\text {fin }}$ \begin{aligned} \left|\chi{A}\right|_{\mathcal{H}{E}}^{2}=& \frac{1}{2} \int{V \times V}\left(\chi_{A}(x)-\chi_{A}(y)\right)^{2} d \rho(x, y) \ =& \frac{1}{2} \int_{V \times V}\left(2 \chi_{A}(x)-2 \chi_{A}(x) \chi_{A}(y)\right) d \rho(x, y) \ =& \int_{V} \int_{V} \chi_{A}(x) d \rho_{x}(y) d \mu(x) \ &-\int_{V} \int_{V} \chi_{A}(x) \chi_{A}(y) d \rho_{x}(y) d \mu(x) \ =& \int_{A} c(x) d \mu(x)-\int_{A} \rho_{x}(A) d \mu(x) \ \leq & \int_{A} c(x) d \mu(x) \end{aligned}

## 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|Spectral theory for graph Laplacians in L2

We will use here the notation introduced in the previous sections. In the next statement, we consider the graph Laplace operator $\Delta$ acting in the Hilbert space $L^{2}(\mu)$. To emphasize this fact, we will use also the notation $\Delta_{2}$. As usual, our basic objects are a measure space $(V, \mathcal{B}, \mu)$ and a symmetric measure $\rho$ such that $\rho_{x}(V)=c(x) \in(0, \infty)$ for $\mu$-a.e. $x \in V$.
Assumption E. We assume in this section that, for every set $A \in \mathcal{B}{\text {fin }}$, the function $$x \mapsto \rho{x}(A)=\int_{V} \chi_{A}(y) d \rho_{x}(y)$$
belongs to $L^{1}(\mu) \cap L^{2}(\mu)$.
Recall that the subspace $\mathcal{D}{\text {fin }}$ is spanned by characteristic functions $\chi{A}$ with $\mu(A)<\infty$. Clearly, $\mathcal{D}_{\text {fin }}$ is dense in $L^{2}(\mu)$.

We use Assumption to justify the definition of the graph Laplace operator $\Delta$ as an unbounded linear operator acting in $L^{2}(\mu)$.
Lemma 7.1. Let
$$\Delta(f)(x)=\int_{V}(f(x)-f(y)) d \rho_{x}(y) .$$
Then
$$\mathcal{D}_{\text {fin }} \subset \operatorname{Dom}(\Delta) \cap L^{2}(\mu)$$
and $\Delta$ is a densely defined operator.

## 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|Structure of the energy space

$$\mathcal{D} \text { fin } \subset \mathcal{H}$$

$$\left|\chi_{A}\right|{\mathcal{H} E}^{2} \leq \int A c(x) d \mu(x),$$ 和 $$\left|\chi{A}\right|{\mathcal{H} E}^{2}=\rho\left(A \times A^{c}\right)$$ 在哪里 $A^{c}:=V \backslash A$. 更一般地说，一个有 $$\left\langle\chi A, \chi{B}\right\rangle_{\mathcal{H E}}=\rho((A \cap B) \times V)-\rho(A \times B) \quad=\nu(A \cap B)-\rho(A \times B)$$

$$\chi A \perp \chi_{B} \Longleftrightarrow \rho((A \backslash B) \times B)=\rho\left((A \cap B) \times B^{c}\right) .$$

$$|\chi A|{\mathcal{H E}}^{2}=\frac{1}{2} \int V \times V\left(\chi{A}(x)-\chi_{A}(y)\right)^{2} d \rho(x, y)=\quad \frac{1}{2} \int_{V \times V}\left(2 \chi_{A}(x)-2 \chi_{A}\right.$$

## 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|Spectral theory for graph Laplacians in L2

$$x \mapsto \rho x(A)=\int_{V} \chi_{A}(y) d \rho_{x}(y)$$

$$\mathcal{D}_{\text {fin }} \subset \operatorname{Dom}(\Delta) \cap L^{2}(\mu)$$

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