# 数学代写|傅里叶分析代写Fourier analysis代考|Unitary Operators

## 数学代写|傅里叶分析代写Fourier analysis代考|Unitary Operators

Definition 7.4 An operator $T \in \mathcal{L}(\mathfrak{G})$ is called a unitary operator if it satisfies $T T^=T^ T=I$ (identity operator).

The Fourier transform and the inverse Fourier transform are typical examples of this category. The set of unitary operators in $\mathcal{L}(\mathfrak{5})$ forms a group with respect to the composition of operators. It is called the unitary group.

Theorem 7.7 The following three statements are equivalent for an operator $T \in$ $\mathcal{L}(\mathfrak{S})$
(i) $T$ is a unitary operator:
(ii) $T(\mathfrak{5})=\mathfrak{5}$ and
$$\langle T x, T y\rangle=\langle x, y\rangle$$
for all $x, y \in \mathfrak{5}$.

## 数学代写|傅里叶分析代写Fourier analysis代考|Stochastic Processes of Second Order

Let $(\Omega, \mathcal{E}, P)$ be a probability space, and $T$ a subset of $\mathbb{R} . T$ is usually interpreted as the space of time and, in this chapter, $T$ is assumed to be either $\mathbb{R}$ or $\mathbb{Z}$. A function $X: T \times \Omega \rightarrow \mathbb{C}$ is said to be a stochastic process if the function $\omega \mapsto X(t, \omega)$ is $(\mathcal{E}, \mathcal{B}(\mathbb{C}))$-measurable for any fixed $t \in T$.

The trajectory of $X(t, \omega)$ for a fixed $\omega$, that is, the function $t \mapsto X(t, \omega)$, is called the sample function of this stochastic process.
Let $\mathcal{T}$ be the set of all the finite tuples of elements of $T$, that is
$$\mathcal{T}=\left{\mathbf{t}=\left(t_1, t_2, \cdots, t_n\right) \mid t_j \in T, j=1,2, \cdots, n, n \in \mathbb{N}\right}$$
$X_{\mathbf{t}}(\omega)$ denotes the vector
$$X_{\mathbf{t}}(\omega)=\left(X\left(t_1, \omega\right), X\left(t_2, \omega\right), \cdots, X\left(t_n, \omega\right)\right), \quad \mathbf{t} \in \mathcal{T} .$$
The set function $v_{X_{\mathrm{t}}}: \mathcal{B}\left(\mathbb{C}^n\right) \rightarrow \mathbb{R}$ defined by
$$v_{X_{\mathbf{t}}}(E)=P\left{\omega \in \Omega \mid X_{\mathbf{t}}(\omega) \in E\right}, \quad E \in \mathcal{B}\left(\mathbb{C}^n\right)$$
is a measure on $\mathcal{B}\left(\mathbb{C}^n\right)$, called the distribution of $X_{\mathbf{t}}(\omega)$. The set of all the distributions $\left{v_{X_{\mathrm{t}}} \mid \mathbf{t} \in \mathcal{T}\right}$ is called the system of finite dimensional distributions determined by $X(t, \omega)$.
$\left{v_{X_{\mathbf{t}}} \mid \mathbf{t} \in \mathcal{T}\right}$ is determined by a given stochastic process $X(t, \omega)$. Conversely, assume now that a probability measure $v_{\mathbf{t}}$ on $\left(\mathbb{C}^n, \mathcal{B}\left(\mathbb{C}^n\right)\right)$ is given for each $\mathbf{t}=$ $\left(t_1, t_2, \cdots, t_n\right) \in \mathcal{T}$. The set of all of them is $\left{v_{\mathbf{t}} \mid \mathbf{t} \in \mathcal{T}\right}$. Does there exist a stochastic process, the system of finite dimensional distributions of which is $\left{v_{\mathbf{t}} \mid \mathbf{t} \in \mathcal{T}\right}$ ? The positive answer to this question is given by the famous theorem due to A.N. Kolmogorov. $^4$

# 傅里叶分析代写

## 数学代写|傅里叶分析代写Fourier analysis代考|Unitary Operators

\langle T x, T y\rangle=\langle x, y\rangle
$$对全部 x, y \in 5. ## 数学代写|傅里叶分析代写Fourier analysis代考|Stochastic Processes of Second Order 让 (\Omega, \mathcal{E}, P) 是一个概率空间，并且 T 的一个子集 \mathbb{R} . T 通常被解释为时间空间，在本章 中， T 被假定为 \mathbb{R} 或者 \mathbb{Z}. 个函数 X: T \times \Omega \rightarrow \mathbb{C} 如果函数 \omega \mapsto X(t, \omega) 是 (\mathcal{E}, \mathcal{B}(\mathbb{C})) – 可测量任何固定 t \in T. 的轨迹 X(t, \omega) 对于一个固定的 \omega ，即函数 t \mapsto X(t, \omega) ，称为该随机过程的样本函数。 让 \mathcal{T} 是元素的所有有限元组的集合 T ，那是 \backslash mathcal {T}=\backslash eft \backslash mathbf {t}=\backslash left(t_1, t_2, \backslash cdots, t_n \backslash right) \backslash mid t j j \backslash in T, j=1,2, \backslash c d o t s, n, n \backslash in X_{\mathbf{t}}(\omega) 表示向量$$
X_{\mathbf{t}}(\omega)=\left(X\left(t_1, \omega\right), X\left(t_2, \omega\right), \cdots, X\left(t_n, \omega\right)\right), \quad \mathbf{t} \in \mathcal{T}


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