MY-ASSIGNMENTEXPERT™可以为您提供sydney MATH4069 Fourier analysis傅里叶分析的代写代考和辅导服务！
这是悉尼大学 傅里叶分析课程的代写成功案例。
MATH4069课程简介
Measure theory is the study of fundamental ideas as length, area, volume, arc length and surface area. It is the basis for Lebesgue integration theory used in advanced mathematics ever since its development in about 1900. Measure theory is also a key foundation for modern probability theory. The course starts by establishing the basics of measure theory and the theory of Lebesgue integration, including important results such as Fubini’s Theorem and the Dominated Convergence Theorem which allow us to manipulate integrals. These ideas are applied to Fourier Analysis which leads to results such as the Inversion Formula and Plancherel’s Theorem. The Radon-Nikodyn Theorem provides a representation of measures in terms of a density. Key ideas of this theory are applied in detail to probability theory to provide a rigorous framework for probability which takes in and generalizes familiar ideas such as distributions and conditional expectation. When you complete this unit you will have acquired a new generalized way of thinking about key mathematical concepts such as length, area, integration and probability. This will give you a powerful set of intellectual tools and equip you for further study in mathematics and probability.
Prerequisites
At the completion of this unit, you should be able to:
- LO1. explain and apply the fundamentals of abstract measure and integration theory
- LO2. explain what an outer measure and use outer measures to construct other measures. Apply these concepts to Lesbesgue measures and other related measures.
- LO3. explain and apply the limit theorems including the dominated convergence theorem and theorems on continuity and differentiability of parameter integrals.
- LO4. explain the properties of Lp spaces
- LO5. use inequalities such as Holder’s, Minkowsi’s, Jensen’s and Young’s inequalities to solve problems
- LO6. explain and apply the properties of the Fourier series on normed spaces
- LO7. generate the measure theoretic foundations of probability theory
- LO8. recall and explain the definition and basic properties of conditional expectation
- LO9. create proofs and apply measure theory in diverse applications.
MATH4069 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）
Suppose $f$ and $f_n$ are measurable functions on $(X, \mu)$. Prove that
(a) $d_f$ is right continuous on $[0, \infty)$.
(b) If $|f| \leq \liminf {n \rightarrow \infty}\left|f_n\right| \mu$-a.e., then $d_f \leq \liminf {n \rightarrow \infty} d_{f_n}$.
(c) If $\left|f_n\right| \uparrow|f|$, then $d_{f_n} \uparrow d_f$.
[Hint: Part (a): Let $t_n$ be a decreasing sequence of positive numbers that tends to zero. Show that $d_f\left(\alpha_0+t_n\right) \uparrow d_f\left(\alpha_0\right)$ using a convergence theorem. Part (b): Let $E={x \in X:|f(x)|>\alpha}$ and $E_n=\left{x \in X:\left|f_n(x)\right|>\alpha\right}$. Use that $\mu\left(\bigcap_{n=m}^{\infty} E_n\right) \leq$ $\liminf {n \rightarrow \infty} \mu\left(E_n\right)$ and $E \subseteq \bigcup{m=1}^{\infty} \bigcap_{n=m}^{\infty} E_n \mu$-a.e. $]$
.(Hölder’s inequality) Let $00$ almost everywhere, let $|g|_{L^r}=\left|g^{-1}\right|_{L^{|r|}}^{-1}$. Show that for $f \geq 0, g>0$ a.e. we have
$$
|f g|_{L^1} \geq|f|_{L^q}|g|_{L^{q^{\prime}}}
$$
(a) Show that the product $f_1 \cdots f_k$ is in $L^p$ and that
$$
\left|f_1 \cdots f_k\right|_{L^p} \leq\left|f_1\right|_{L^{p_1}} \cdots\left|f_k\right|_{L^{p_k}} .
$$
(b) When no $p_j$ is infinite, show that if equality holds in part (a), then it must be the case that $c_1\left|f_1\right|^{p_1}=\cdots=c_k\left|f_k\right|^{p_k}$ a.e. for some $c_j \geq 0$.
(c) Let $00$ almost everywhere, let $|g|_{L^r}=\left|g^{-1}\right|_{L^{|r|}}^{-1}$. Show that for $f \geq 0, g>0$ a.e. we have
$$
|f g|_{L^1} \geq|f|_{L^q}|g|_{L^{q^{\prime}}}
$$
Let $(X, \mu)$ be a measure space.
(a) If $f$ is in $L^{p_0}(X, \mu)$ for some $p_0<\infty$, prove that
$$
\lim {p \rightarrow \infty}|f|{L^p}=|f|_{L^{\infty}} .
$$
(b) (Jensen’s inequality) Suppose that $\mu(X)=1$. Show that
$$
|f|_{L^p} \geq \exp \left(\int_X \log |f(x)| d \mu(x)\right)
$$
for all $00$, then
$$
\lim {p \rightarrow 0}|f|{L^p}=\exp \left(\int_X \log |f(x)| d \mu(x)\right)
$$
with the interpretation $e^{-\infty}=0$.
[Hint: Part (a): Given $0<\varepsilon<|f|_{L^{\infty}}$, find a measurable set $E \subseteq X$ of positive measure such that $|f(x)| \geq|f|_{L^{\infty}}-\varepsilon$ for all $x \in E$. Then $|f|_{L^p} \geq\left(|f|_{L^{\infty}}-\varepsilon\right) \mu(E)^{1 / p}$ and thus $\liminf {p \rightarrow \infty}|f|{L^p} \geq|f|_{L^{\infty}}-\varepsilon$. Part (b) is a direct consequence of Jensen’s inequality $\int_X \log |h| d \mu \leq \log \left(\int_X|h| d \mu\right)$. Part (c): Fix a sequence $00$. The Lebesgue monotone convergence theorem yields $\int_X h_n d \mu \uparrow \int_X h d \mu$, hence $\int_X \frac{1}{p_n}\left(|f|^{p_n}-1\right) d \mu \downarrow \int_X \log |f| d \mu$, where the latter could be $-\infty$. Use
$$
\exp \left(\int_X \log |f| d \mu\right) \leq\left(\int_X|f|^{p_n} d \mu\right)^{\frac{1}{p_n}} \leq \exp \left(\int_X \frac{1}{p_n}\left(|f|^{p_n}-1\right) d \mu\right)
$$
to complete the proof.]
Let $\left{f_j\right}_{j=1}^N$ be a sequence of $L^p(X, \mu)$ functions.
(a) (Minkowski’s inequality) For $1 \leq p \leq \infty$ show that
$$
\left|\sum_{j=1}^N f_j\right|_{L^p} \leq \sum_{j=1}^N\left|f_j\right|_{L^p} .
$$
(b) (Minkowski’s inequality) For $0<p<1$ and $f_j \geq 0$ prove that
$$
\sum_{j=1}^N\left|f_j\right|_{L^p} \leq\left|\sum_{j=1}^N f_j\right|_{L^p}
$$
(c) For $0<p<1$ show that
$$
\left|\sum_{j=1}^N f_j\right|_{L^p} \leq N^{\frac{1-p}{p}} \sum_{j=1}^N\left|f_j\right|_{L^p} .
$$
(d) The constant $N^{\frac{1-p}{p}}$ in part (c) is best possible.
Hint: Part (c): Use Exercise 1.1.4(c). Part (d): Take $\left{f_j\right}_{j=1}^N$ to be characteristic functions of disjoint sets with the same measure.
MY-ASSIGNMENTEXPERT™可以为您提供STANFORD EE261 FOURIER ANALYSIS傅里叶分析的代写代考和辅导服务！
