MY-ASSIGNMENTEXPERT™可以为您提供mathweb.ucsd.edu MTH710 Fourier analysis 傅里叶分析的代写代考和辅导服务！
MTH710课程简介
Textbook: The main source we will follow are Bruce Driver’s excellent Probability notes:
Probability Tools with Examples, by Bruce Driver
Here are a few other textbooks we recommend as auxiliary sources; all are freely available to UCSD personnel.
Probability: Theory and Examples (5th Edition), by Rick Durrett
A Probability Path by Sidney Resnick
AModern Approach to Probability Theory by Bert Fristedt and Lawrence Gray
Probability Theory: A Comprehensive Course by Achim Klenke
If you are not on UCSD campus, make sure you are logged into the VPN in order to gain access; you can find instructions on how to do this here.
Prerequisites
Weekly Contact: Lecture: 3 hrs.GPA Weight: 1.00Course Count: 1.00Billing Units: 1
Prerequisites
MTH 108 or MTH 141 and MTH 231 or MTH 310 or MTH 240
Co-Requisites
None
Antirequisites
None
Custom Requisites
None
MTH710 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）
Let us define
$$
J_0(x)=\frac{1}{2 \pi} \int_0^{2 \pi} \cos (x \sin (\theta)) d \theta
$$
Show that
$$
f(x) \mapsto F(\xi)=2 \pi \int J_0(2 \pi \xi x) f(x) x d x
$$
defines a unitary map from $L^2([0, \infty), r d r)$ to itself. Describe the relation to the Fourier transform of radial functions in two dimensions.
To show that the map defined by $f(x) \mapsto F(\xi) = 2\pi \int J_0(2\pi \xi x) f(x) x dx$ is unitary, we need to show that it is an isometry, i.e., it preserves the inner product. Let $f,g \in L^2([0,\infty), r dr)$, then we have
\begin{align*} \langle f, g \rangle &= \int_0^\infty f(x) \overline{g(x)} r dx \ &= \int_0^\infty f(x) \overline{g(x)} \frac{x}{x} r dr \ &= \int_0^\infty f(x) \overline{g(x)} \frac{x}{\sqrt{x}} \sqrt{x} r dr \ &= \int_0^\infty f(x) \overline{g(x)} \frac{x}{\sqrt{x}} \sqrt{x} r^{\frac{3}{2}} \frac{1}{r} dr \ &= \int_0^\infty f(x) \overline{g(x)} \sqrt{x} J_{\frac{1}{2}}(2\pi x r) r dr &\text{(using the expression for the Fourier-Bessel transform)}\ &= \int_0^\infty F(\xi) \overline{G(\xi)} d\xi &\text{(using the expression for $F(\xi)$)}\ &= \langle F,G\rangle, \end{align*}
where we have used the expression for the Fourier-Bessel transform of radial functions $\$ \mathrm{f} \$$ in two dimensions:
$$
F(\xi)=\int_0^{\infty} f(x) J_0(2 \pi \xi x) x d x
$$
Therefore, the map is an isometry and it preserves the inner product, so it is unitary.
The function $J_0(x)$ is the zeroth order Bessel function of the first kind, and it plays an important role in the Fourier analysis of radial functions in two dimensions. The expression $J_0(x) = \frac{1}{2\pi} \int_0^{2\pi} \cos(x\sin\theta) d\theta$ is known as the Poisson kernel, and it represents the Fourier coefficients of the two-dimensional radial Dirac delta function $\delta_0(r)$. That is, if we define $f(r) = \delta_0(r)$, then $F(\xi) = J_0(2\pi \xi)$, which gives us the Fourier transform of the radial Dirac delta function.
More generally, if $\$ \mathrm{f} \$$ is a radial function in two dimensions (i.e., it depends only on $\left.\$ r=\backslash s q r t\left{x^{\wedge} 2+y^{\wedge} 2\right} \$\right)$, then its Fourier transform can be expressed in terms of the Fourier-Bessel transform as
$$
\hat{f}(\xi)=2 \pi \int_0^{\infty} J_0(2 \pi \xi r) f(r) r d r .
$$
This expresses the Fourier transform of a radial function as an integral involving the Bessel function $J_0$, which arises due to the rotational symmetry of the function.
Let $d \rho$ be a probability measure on $\mathbb{R}$ with $\int x d \mu(x)=0$ and $\int x^4 d \rho(x)<\infty$. Prove the central limit theorem for the sum of independent random variables with this distribution.
Specifically, if $X_1, X_2, \ldots$ are $d \rho$-distributed, show that for any Schwartz function $f$
$$
\mathbb{E}\left{f\left(\frac{X_1+\cdots+X_n}{n^{1 / 2}}\right)\right} \longrightarrow \frac{1}{\sqrt{2 \pi \sigma^2}} \int \exp \left{-\frac{x^2}{2 \sigma^2}\right} f(x) d x
$$
as $n \rightarrow \infty$. Hint: first show convergence for $f(x)=e^{-2 \pi i x \xi}$ uniformly for $\xi$ in a compact set.
Let $\$ S_{-} n=\backslash f r a c\left{X _1+X _2+\backslash\right.$ cdots $\left.+X _n\right}{\backslash s q r t{n}} \$$. We want to show that for any Schwartz function $\$ \mathbb{F} \$$, we have
$$
\mathbb{E}\left{f\left(S_n\right)\right} \longrightarrow \frac{1}{\sqrt{2 \pi \sigma^2}} \int_{-\infty}^{\infty} e^{-\frac{x^2}{2 \sigma^2}} f(x) d x
$$
as $n \rightarrow \infty$, where $\sigma^2 = \operatorname{Var}(X_1)$.
First, we will show the convergence for the specific Schwartz function $f(x) = e^{-2\pi i x \xi}$, where $\xi$ is a fixed parameter. We have \begin{align*} \mathbb{E}\left{f(S_n)\right} &= \mathbb{E}\left{e^{-2\pi i \xi S_n}\right} \ &= \mathbb{E}\left{\cos(2\pi \xi S_n)\right} – i\mathbb{E}\left{\sin(2\pi \xi S_n)\right} \ &= \int_{-\infty}^\infty \cos(2\pi \xi x) dF_n(x) – i\int_{-\infty}^\infty \sin(2\pi \xi x) dF_n(x), \end{align*} where $F_n$ is the empirical distribution function of $S_n$, given by $F_n(x) = \frac{1}{n}\sum_{k=1}^n \mathbf{1}_{{S_k \leq x}}$.
Now, we can use the characteristic function of $\$ X _1 \$$ to write
$$
\mathbb{E}\left{e^{2 \pi i \xi X_1}\right}=\int_{-\infty}^{\infty} e^{2 \pi i \xi x} d \rho(x)=1
$$
which implies that
$$
\mathbb{E}\left{\cos \left(2 \pi \xi X_1\right)\right}=\mathbb{E}\left{\sin \left(2 \pi \xi X_1\right)\right}=\frac{1}{2}
$$
Using the Lindeberg-Feller Central Limit Theorem, we can show that for any \$\epsilon > 0 \$, we have
$$
\sup {x \in \mathbb{R}}\left|F_n(x)-\int{-\infty}^x d \rho(y)\right| \longrightarrow 0 \quad \text { as } n \rightarrow \infty
$$
and therefore, we have \begin{align*} \left|\mathbb{E}\left{f(S_n)\right} – \int_{-\infty}^\infty e^{-2\pi^2\xi^2\sigma^2} e^{-2\pi i \xi x} \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{x^2}{2\sigma^2}} dx \right| &\leq \mathbb{E}\left{\left|\cos(2\pi \xi S_n) – \frac{1}{2}\right|\right} + \mathbb{E}\left{\left|\sin(2\
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