傅里叶分析代写|6CCM318A Fourier Analysis

Let $f: S^{1} \rightarrow \mathbb{C}$ be an integrable function.
(a) Define the Abel sums $\left(A_{r}(f)\right)(\cdot)$ related to $f, r>0$, and the Poisson kernel $P_{r}(\theta)$. State without a proof whether $\left{P_{r}(\cdot)\right}_{0 \leq r<1}$ is a family of good kernels as $r \rightarrow 1-$.
$[3$ marks $]$
(b) Prove that the Fourier series of $f$ is Abel convergent to $f$ at all points of continuity $\theta \in S^{1}$ of $f$. You may use the Good Kernels theorem, provided you state it correctly.
$[7$ marks
(c) For each of the following statements determine whether it is true or false. Justify your answers with a proof or a counterexample.
(i) If $f$ is continuous everywhere on $S^{1}$, then the Abel convergence of the Fourier series of $f$ is uniform.
(ii) The Fourier series of $f$ is Abel convergent to $f$ everywhere.
(iii) If a series of numbers $\sum_{n=1}^{\infty} c_{n}$ is Abel convergent to $s$, then it is also convergent to $s$ in the usual sense.
(iv) If the Fourier series of $f$ converges uniformly to $f$, then it is also uniformly Abel convergent to $f$.
$[15 \mathrm{ma}$

T

(a) For Schwartz functions $f, g \in \mathcal{S}(\mathbb{R})$ define the convolution $(f * g)(\cdot)$ and state without a proof whether $f * g \in \mathcal{S}(\mathbb{R})$. Does $f * g(\cdot)$ make sense?
$[3$ marks $]$
(b) Prove that if $f, g \in \mathcal{S}(\mathbb{R})$ are Schwartz functions, then
$$
\widehat{f * g}(\xi)=\widehat{f}(\xi) \cdot \widehat{g}(\xi)
$$
$[7$ marks $]$
(c) For each of the following statements determine whether it is true or false. Justify your answers with a proof or a counterexample.
(i) $f(x)=e^{-|x|}$ is a Schwartz function.
(ii) $f(x)=e^{-x^{4}}$ is a Schwartz function.
(iii) If $f, g \in \mathcal{S}(\mathbb{R})$ are Schwartz functions, then
$$
\left(\mathcal{F}^{} f\right) \cdot\left(\mathcal{F}^{} g\right)=\mathcal{F}^{*}(f * g)
$$
(iv) If $f, g \in \mathcal{S}(\mathbb{R})$ are Schwartz functions, then $f \cdot g \in \mathcal{S}(\mathbb{R})$.
(v) If $f, g \in \mathcal{S}(\mathbb{R})$ are Schwartz functions so that for all $x \in \mathbb{R}$, $g(x)>0$, then $f(x) / g(x) \in \mathcal{S}(\mathbb{R})$.
marks]

Let $f: S^{1} \rightarrow \mathbb{C}, f(\theta)=|\theta|$ on $\theta \in[-\pi, \pi]$.
(a) Evaluate the Fourier coefficients $\widehat{f}(n)$ of $f$ and hence find the Fourier series of $f$ in its complex and its real forms.
$[15$ marks
(b) Use your answers to part (a) to determine whether the Fourier series of $f$ converges to $f$, and whether it converges uniformly to $f$ w.r.t. $x \in S^{1}$.
$[5$ marks $]$
(c) By computing the Fourier coefficients of $g:=f * f$ find a maximum possible $k \geq 0$ so that $g \in C^{k}\left(S^{1}\right)$.
$[5$ marks $]$

Let $f: \mathbb{R} \rightarrow \mathbb{C}$ be the function
$$
f(x)=\chi_{[-1,1]}(x)=\left{\begin{array}{ll}
1 & x \in[-1,1] \
0 & \text { otherwise }
\end{array}\right.
$$
i.e., the characteristic function of the unit interval in $\mathbb{R}$.
(a) Evaluate the Fourier transform of $f$, and use your result to compute the Fourier transform of the function $f(x)=\frac{\sin x}{x}$. $[12 \mathrm{marks}]$
(b) Find the convolution $g(x):=(f * f)(x)$ of $f$ with itself, and its Fourier transform $\widehat{f}(\xi)$. (Hint: you are not required to compute the Fourier transform of $g$ directly.)
$[7$ marks
(c) By applying the Plancherel identity or otherwise, evaluate the integral $\int_{0}^{\infty} \frac{\sin (x)^{2}}{x^{2}} d x$
marks]