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|Definition and properties of the finite energy space HE

We consider a class of Borel functions over $(V, \mathcal{B}, \mu)$ which is formed by functions of finite energy. In other words, this section is focused on a measurable analogue of the energy Hilbert space which was extensively studied in the context of discrete networks, see, e.g., [Cho14, JP16, Jor12, LP16].

Definition 6.6. Let $(V, \mathcal{B}, \mu)$ be a standard measure space with $\sigma$-finite measure $\mu$. Suppose that $\rho$ is a symmetric measure on the Cartesian product $(V \times V, \mathcal{B} \times \mathcal{B})$. We say that a Borel function $f: V \rightarrow \mathbb{R}$ belongs to the finite energy space $\mathcal{H}{E}=\mathcal{H}$ if $$\iint{V \times V}(f(x)-f(y))^{2} d \rho(x, y)<\infty .$$
If the measure $\rho$ is defined in terms of a conductance function $c_{x y}$, then a function $f$ is in $\mathcal{H}$ when
$$\int_{V}\left(\int_{V} c_{x y}(f(x)-f(y))^{2} d \mu(y)\right) d \mu(x)<\infty .$$
Remark 6.7. (1) It follows from the Cauchy-Schwarz inequality in the space $L^{2}(\rho)$ that the set $\mathcal{H}$ is a vector space. It contains all constant function $k$. Since for the functions $f$ and $f+k$, the quantity in (6.2) is the same, we can identify such functions in the space $\mathcal{H}$. That is $\mathcal{H}$ can be treated as the space of classes of equivalent functions where $f \sim g$ iff $f-g$ is a constant. With some abuse of notation we will denote this quotient space again by $\mathcal{H}$. We show below that $\mathcal{H}$ is a Hilbert space.

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|Energy space is embedded into dissipation space

Let $P$ be a Markov operator and $x$ is a fixed point in $V$. Denote by $P(x, A)$ the probability measure defined by $P$ as in Section 4 . This means that
$$P(x, f)=\int_{V} f(y) d \bar{\rho}{x}(y)=P(f)(x)$$ where $f$ is a Borel function. If $X{n}(\omega)$ is a corresponding sequence of random variables on $\Omega_{x}$, then we have the following formulas for the conditional expectation $\mathbb{E}{x}$ with respect to the probability measure $\mathbb{P}{x}$ :
\begin{aligned} &\mathbb{E}{x}\left(f \circ X{0}\right)=\int_{\Omega_{x}} f\left(X_{0}(\omega)\right) d \mathbb{P}{x}(\omega)=\int{\Omega_{x}} f(x) d \mathbb{P}{x}(\omega)=f(x), \ &\mathbb{E}{x}\left(f \circ X_{1}\right)=\int_{\Omega_{x}} f\left(X_{1}(\omega)\right) d \mathbb{P}{x}(\omega)=\int{V} f(y) P(x, d y)=P(f)(x) \end{aligned}
where $y=X_{1}(\omega)$.
Definition 6.11. Define a linear operator $\partial: \mathcal{H}{E} \rightarrow$ Diss by the formula: $$\partial: f \mapsto f \circ X{1}-f \circ X_{0} .$$
Similarly, we set
$$\partial_{n}: f \mapsto f \circ X_{n+1}-f \circ X_{n} .$$
Remark that we use the same notation $\partial$ as in (6.6) of Lemma $6.8$ because these operators are essentially similar.
Lemma 6.12. The operator $\partial: \mathcal{H}_{E} \rightarrow$ Diss defined in (6.9) is an isometry.

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|Definition and properties of the finite energy space HE

$$\iint V \times V(f(x)-f(y))^{2} d \rho(x, y)<\infty .$$

$$\int_{V}\left(\int_{V} c_{x y}(f(x)-f(y))^{2} d \mu(y)\right) d \mu(x)<\infty .$$

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|Energy space is embedded into dissipation space

$$P(x, f)=\int_{V} f(y) d \bar{\rho} x(y)=P(f)(x)$$

$$\mathbb{E} x(f \circ X 0)=\int_{\Omega_{x}} f\left(X_{0}(\omega)\right) d \mathbb{P} x(\omega)=\int \Omega_{x} f(x) d \mathbb{P} x(\omega)=f(x)$$

$$\partial: f \mapsto f \circ X 1-f \circ X_{0} .$$

$$\partial_{n}: f \mapsto f \circ X_{n+1}-f \circ X_{n} .$$

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