MY-ASSIGNMENTEXPERT™可以为您提供 newcastle MATH3205 Fourier analysis傅里叶分析的代写代考和辅导服务！
这是纽卡斯尔大学 傅里叶分析课程的代写成功案例。
MATH3205课程简介
Introduces the basics of Fourier analysis as a prelude to applications. The course develops Fourier analysis from a general pure mathematical perspective starting with Lebesgue integration and elements of the theory of Hilbert spaces, leading to Fourier series, Fourier integrals and the fast Fourier transform, and then to applications such as partial differential equations and sampling. These subjects are of great importance to the electrical engineering and physics communities. The course concludes with more modern topics such as Gabor and wavelet transforms.
Prerequisites
On successful completion of the course students will be able to:
1. In-depth knowledge of Fourier analysis and its applications to problems in physics and electrical engineering.
2. An ability to communicate reasoned arguments of a mathematical nature in both written and oral form.
3. An ability to read and construct rigorous mathematical arguments.
MATH3205 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）
For a function $f$ on a locally compact group $G$ and $t \in G$, let ${ }^t f(x)=f(t x)$ and $f^t(x)=f(x t)$. Show that
$$
{ }^t f * g={ }^t(f * g) \quad \text { and } \quad f * g^t=(f * g)^t
$$
whenever $f, g \in L^1(G)$, equipped with left Haar measure.
Let $G$ be a locally compact group with left Haar measure. Let $f \in L^p(G)$ and $\tilde{g} \in L^{p^{\prime}}(G)$, where $10$ there exists a relatively compact symmetric neighborhood of the origin $U$ such that $u \in U$ implies $|u \widetilde{g}-\widetilde{g}|_{L^{p^{\prime}(G)}}<\varepsilon$ and therefore
$$
|(f * g)(v)-(f * g)(w)|<|f|_{L^p} \varepsilon
$$
whenever $v w^{-1} \in U$.
Let $G$ be a locally compact group and let $1 \leq p \leq \infty$. Let $f \in L^p(G)$ and $\mu$ be a finite Borel measure on $G$ with total variation $|\mu|$. Define
$$
(\mu * f)(x)=\int_G f\left(y^{-1} x\right) d \mu(y) .
$$
Show that if $\mu$ is an absolutely continuous measure, then the preceding definition extends (1.2.4). Prove that $|\mu * f|_{L^p(G)} \leq|\mu||f|_{L^p(G)}$.
Show that a Haar measure $\lambda$ for the multiplicative group of all positive real numbers is
$$
\lambda(A)=\int_0^{\infty} \chi_A(t) \frac{d t}{t} .
$$
Let $G=\mathbf{R}^2 \backslash{(0, y): y \in \mathbf{R}}$ with group operation $(x, y)(z, w)=(x z, x w+y)$. [Think of $G$ as the group of all $2 \times 2$ matrices with bottom row $(0,1)$ and nonzero top left entry.] Show that a left Haar measure on $G$ is
$$
\lambda(A)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \chi_A(x, y) \frac{d x d y}{x^2},
$$
while a right Haar measure on $G$ is
$$
\rho(A)=\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \chi_A(x, y) \frac{d x d y}{|x|} .
$$
MY-ASSIGNMENTEXPERT™可以为您提供 newcastle MATH3205 Fourier analysis傅里叶分析的代写代考和辅导服务！
