0$ and $\kappa(n)\rightarrow \infty$ as $n\rightarrow\infty$, show that we can find $n(j)\rightarrow\infty$ such that $\sum_{j=1}^{\infty}2^{j}/\kappa(n(j))$ converges. Deduce that we can find a continuous function $f$ such that $\limsup_{n\rightarrow\infty}\kappa(n)\hat{f}(n)=\infty$. \end{question} \begin{question} Show that \[\frac{\card\{1\leq n\leq N \mid \langle \log_{10}n\rangle \in [0,1/2]\}}{N}\] does not tend to limit as $N\rightarrow\infty$. Show however that given any $\epsilon>0$ and any $x\in[0,1]$ we can find a positive integer $n$ such that \[|\langle \log_{10}n\rangle-x|<\epsilon\] as $N\rightarrow\infty$. Prove the same results with $\log_{10}$ replaced by $\log_{e}$. \end{question} \begin{question} Using the kind of ideas behind the proof of of Weyl's theorem (Theorem~\ref{Weyl}), or otherwise, prove the following results. (i) If $f:{\mathbb T}\rightarrow{\mathbb C}$ is continuous, then $\hat{f}(n)\rightarrow 0$ as $|n|\rightarrow\infty$. (This is a version of the Lebesgue--Riemann lemma.) (ii) If $f:{\mathbb T}\rightarrow{\mathbb R}$ is continuous, then \[\int_{0}^{2\pi}f(t)|\sin nt|\,dt \rightarrow \frac{2}{\pi}\int_{0}^{2\pi}f(t)\,dt\] as $n\rightarrow\infty$. \end{question} \begin{question} Let $R$ be a rectangle cut up into smaller rectangles $R(1)$, $R(2)$, \dots, $R(k)$ each of which has sides parallel to the sides of $R$. Then, if each $R(j)$ has at least one pair of sides of integer length, it follows that $R$ has at least one pair of sides of integer length. First try and prove this without using Fourier analysis. Then try and prove the result using what is, in effect, a Fourier transform \[\iint_{R(j)}\exp\big(2\pi i(x+y)\big)\,dx\,dy.\] \end{question} \begin{question}If $f,\ g:{\mathbb T}\rightarrow{\mathbb C}$ are well behaved, let us define their \emph{convolution} $f*g:{\mathbb T}\rightarrow{\mathbb C}$ by \[f*g(t)=\frac{1}{2\pi}\int_{\mathbb T}f(t-s)g(s)\,ds.\] Show that $\widehat{f*g}(n)=\hat{f}(n)\hat{g}(n)$. Show that if $P$ is a trigonometric polynomial $P*f$ is a trigonometric polynomial. Identify $D_{N}*f$ and $K_{N}*f$ where $D_{N}$ and $K_{N}$ are the Dirichlet and Fej\'{e}r kernels. Suppose that $L_{N}(t)=A_{N}K_{N}^{2}(t)$ with $A_{N}$ chosen so that $\frac{1}{2\pi}\int_{\mathbb T}L_{N}(t)\,dt=1$. Show that, if $f$ is continuous, $L_{N}*f(t)\rightarrow f(t)$. \end{question} \begin{question}\label{E, Liouville}. Let $E(x)=(2\pi)^{-1/2}\exp(-x^{2}/2)$. Show, by changing to polar coordinates, that \begin{align*} \left(\int_{-\infty}^{\infty}E(x)\,dx\right)^{2} &=\int_{-\infty}^{\infty}E(x)\,dx\int_{-\infty}^{\infty}E(y)\,dy\\ &=\frac{1}{2\pi}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \exp(-(x^{2}+y^{2}))\,dx\,dy\\ &=\int_{0}^{\infty}r\exp(-r^{2}/2)\,dr=1. \end{align*} Kelvin once asked his class if they knew what a mathematician was. He wrote the formula \[\int_{\infty}^{\infty}e^{-x^{2}/2}\,dx=\sqrt{\pi}\] and the board and said. `A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician.' \end{question} \begin{question}\label{E, differentiable everywhere} (i) Explain why \[\sum_{j=-N}^{N}|a_{j}b_{j}|\leq \left(\sum_{j=-N}^{N}|a_{j}|^{2}\right)^{1/2} \left(\sum_{j=-N}^{N}|b_{j}|^{2}\right)^{1/2}\] for all $a_{j},\,b_{j}\in{\mathbb C}$. (ii) Use~(i) to show that, if $\sum_{j=-\infty}^{\infty}|a_{j}|^{2}$ and $\sum_{j=-\infty}^{\infty}|b_{j}|^{2}$ converges, then $\sum_{j=-\infty}^{\infty}|a_{j}b_{j}|$ converges and \[\sum_{j=-\infty}^{\infty}|a_{j}b_{j}|\leq \left(\sum_{j=-\infty}^{\infty}|a_{j}|^{2}\right)^{1/2} \left(\sum_{j=-\infty}^{\infty}|b_{j}|^{2}\right)^{1/2}.\] (iii) If $f:{\mathbb T}\rightarrow{\mathbb C}$ is continuously differentiable, explain why \[\sum_{j=-\infty}^{\infty}j^{2}|\hat{f}(j)|^{2} \leq \frac{1}{2\pi}\int_{\mathbb T}|f'(t)|^{2}\,dt.\] (iv) Use~(ii) to show that $\sum_{j=-\infty}^{\infty}|\hat{f}(j)|$ converges. Deduce that $\sum_{j=-\infty}^{\infty}\hat{f}(j)\exp ijt$ converges uniformly to $f(t)$. \end{question} \begin{question} (i) If $u:{\mathbb T}\rightarrow{\mathbb R}$ is once continuously differentiable and $\frac{1}{2\pi}\int_{\mathbb T}u(t)\,dt=0$, show that \[\frac{1}{2\pi}\int_{\mathbb T}(u(t))^{2}\,dt \leq \frac{1}{2\pi}\int_{\mathbb T}(u'(t))^{2}\,dt\] with equality if and only if $u(t)=C\cos(t+\phi)$ for some constants $C$ and $\phi$. (ii) Use~(i) to show that, if $v:[0,\pi/2]\rightarrow{\mathbb R}$ is once continuously differentiable with $v(0)=0$ and $v'(\pi/2)=0$ then \[\int_{0}^{\pi/2}(v(t))^{2}\,dt \leq \int_{0}^{\pi/2}(v'(t))^{2}\,dt\] with equality if and only if $u(t)=C\sin t$ for some constant $C$. (iii) By approximating $w$ by functions of the type considered in~(iii) show that, if $w:[0,\pi/2]\rightarrow{\mathbb R}$ is once continuously differentiable with $w(0)=0$, then \[\int_{0}^{\pi/2}(w(t))^{2}\,dt \leq \int_{0}^{\pi/2}(w'(t))^{2}\,dt.\] (This is Wirtinger's inequality.) \end{question} \begin{question} (i) By applying Poisson's formula to the function $f$ defined by $f(x)=\exp(-t|x|/2\pi)$ show that \[2(1-e^{-t})^{-1} =\sum_{n=-\infty}^{\infty}2t(t^{2}+4\pi^{2}n^{2})^{-1}.\] (ii) By expanding $(t^{2}+4\pi n^{2})^{-1}$ and interchanging sums (justifying this, if you can, just interchanging, if not) deduce that \[2(1-e^{-t})^{-1}=1+2t^{-1}+\sum_{m=0}^{\infty}c_{m}t^{m}\] where $c_{2m}=0$ and \[c_{2m+1}=a_{2m+1}\sum_{n=1}^{\infty}n^{-2m}\] for some value of $a_{2m+1}$ to be given explicitly. (iii) Hence obtain Euler's formula \[\sum_{n=1}^{\infty}n^{-2m}= (-1)^{m-1}2^{2m-1}b_{2m-1}\pi^{2m}/(2m-1)!\] for $m\geq 1$, where the $b_{m}$ are defined by the formula \[(e^{y}-1)^{-1}=y^{-1}-2^{-1}+\sum_{n=1}^{\infty}b_{n}y^{n}/n!\] (The $b_{n}$ are called Bernoulli numbers.) \end{question} \begin{question}{\bf (The Gibbs Phenomenon.)}\label{E, Gibbs} Ideally you should first look at what happens when we try to reconstruct a reasonable discontinuous function from its Fourier sums and then use this question to explain what you see. There are a number of questions linked to this one but you need not do them to understand what is going on. We have only discussed Fourier series for continuous functions in this course. It is possible to use what we already know to discuss `well behaved' discontinuous functions. Let $F:{\mathbb T}\rightarrow{\mathbb R}$ be defined by \begin{equation*} F(t)= \begin{cases} \pi-t&\text{for $0\leq t\leq\pi$}\\ 0&\text{for $t=0$}\\ \-pi-t&\text{for $-\pi0$, $I(0)=0$, $I(t)=-L$ for $t<0$. (iv) Find a continuous function $g:[0,\pi]\rightarrow{\mathbb R}$ such that $g(t)\geq 0$ for all $t\in[0,\pi]$, $g(\pi/2)>0$ and \[\left|\frac{\sin x}{x}\right|\geq \frac{g(x-n\pi)}{n}\] for all $n\pi\leq x\leq(n+1)\pi$ and all integer $n\geq 1$. Hence, or otherwise, show that $\int_{0}^{\infty}|(\sin x)/x|\,dx$ fails to converge. \end{question} \begin{question}\label{Q, infinite Dirichlet 2} Although the existence of the infinite integral $\int_{0}^{\infty}\frac{\sin x}{x}\,dx$ is very important, its actual value is less important. It is, however, reasonably easy to find using our knowledge of the Dirichlet kernel, in particular the fact that \[2\pi=\int_{-\pi}^{\pi}\frac{\sin\big((n+\tfrac{1}{2})x\big)} {\sin \tfrac{x}{2}}\,dx\] (see Lemma~\ref{L, Dirichlet's kernel}~(iii)). (i) If $\epsilon>0$, show that \[\int_{-\epsilon}^{\epsilon}\frac{\sin \lambda x}{x}\,dx \rightarrow \int_{-\infty}^{\infty}\frac{\sin x}{x}\,dx,\] as $\lambda\rightarrow\infty$. (ii) If $\pi\geq \epsilon>0$, show, by using the estimates from the alternating series test, or otherwise, that \[\int_{-\epsilon}^{\epsilon} \frac{\sin\big((n+\tfrac{1}{2})x\big)} {\sin \tfrac{x}{2}}\,dx\rightarrow \int_{-\pi}^{\pi}\frac{\sin\big((n+\tfrac{1}{2})x\big)} {\sin \tfrac{x}{2}}\,dx=2\pi\] as $n\rightarrow\infty$. (iii) Show that \[\left|\frac{2}{x}-\frac{1}{\sin \tfrac{1}{2}x}\right|\rightarrow 0\] as $x\rightarrow 0$. and deduce that \[\int_{0}^{\infty}\frac{\sin x}{x}\,dx=\frac{\pi}{2}.\] \end{question} \begin{question}\label{E, Fourier} This question refers back to Question~\ref{E, Gibbs}. There we discussed the behaviour of $S_{n}(F,t)$ when $t$ is small but did not show that $S_{n}(F,t)$ behaves well when $t$ is far from $0$. This follows from general theorems but we shall prove it directly. This brings us into direct contact with Fourier since he used $F$ as a test case for his statement that any function\footnote{We would now say any `reasonable function' but Fourier and his contemporaries had a narrower view of what constituted a function.} had a Fourier expansion. (i) Show that \[S_{n}(F,t)=\int_{0}^{t}\sum_{r=1}^{n}\cos rx\,dx\] and \[\sum_{r=1}^{n}\cos rx=\frac{\sin(n+\tfrac{1}{2})x}{2\sin\tfrac{x}{2}}.\] (ii) Deduce that \[S_{n}(F,t)=2\int_{0}^{t}\frac{\sin(n+\tfrac{1}{2})x}{x}\,dx -t+\int_{0}^{t}g(x)\sin(n+\tfrac{1}{2})x\,dx\] where $g(0)=0$ and \[g(x)=\frac{x-2\sin\tfrac{x}{2}}{x\sin\tfrac{x}{2}}.\] for $0<|x|<\pi$. (iii) Show that $g$ is continuous at $0$. Show that $g$ is differentiable at $0$ and find its derivative there. Show that $g$ is continuously differentiable on $(-T,T)$ for all $0 0$. \noindent If $K>0$, let us write \[E_{n}=\{x\in {\mathbb T}\,:\,h_{n}(x)\geq K\}.\] Show that \[\int_{E_{n}}h_{n}(t)\,dt\rightarrow 1\] as $n\rightarrow\infty$. Deduce that there exists an $N(K)$ such that \[\int_{E_{n}}h_{n}(t)^{2}\,dt\geq \frac{K}{2}\] for all $n\geq N(K)$. If $f:{\mathbb T}\rightarrow{\mathbb R}$ is continuous with $f(0)>0$, deduce that \[\int_{\mathbb T}h_{n}(x)^{2}f(x)\,dx\rightarrow\infty\] as $n\rightarrow \infty$. \end{question} \begin{question}\label{work convolution} (i) By using the mean value theorem or some other appropriate version of Taylors theorem,show that, if $f\in{\mathcal D}$, \[\frac{f(x+h)-f(x)}{h}\rightarrow f'(x)\] uniformly in $x$ as $h\rightarrow 0$. (ii) If $h_{n}\neq 0$, $h_{n}\rightarrow 0$ and $f\in{\mathcal D}$, show that \[\frac{f(x+h_{n})-f(x)}{h_{n}}\arrowD f'(x).\] Deduce that, if $S\in{\mathcal D}'$ and we write \[g(x)=\langle S(s),f(x+s)\rangle\] then \[\frac{g(x+h_{n})-g(x}{h_{n}}\rightarrow\langle S(s),f'(s+x)\rangle\] as $n\rightarrow\infty$. (iii) If $f\in{\mathcal D}$ and $S\in{\mathcal D}'$, show that \[\frac{\langle S(s),f(x+h+s)\rangle-\langle S(s),f(x+s)\rangle}{h} \rightarrow \langle S(s),f'(s+x)\rangle\] as $h\rightarrow 0$. (iv) If $f\in{\mathcal D}$ and $S\in{\mathcal D}'$, show that, if $g(x)=\langle S(s),f(x+s)\rangle$ then $g\in{\mathcal D}$. Deduce that, if $T\in{\mathcal D}'$ $\langle T(u),\langle S(s),f(u+s)\rangle$ is a well defined object. (v) If $T,\,S\in{\mathcal D}'$ we set \[\langle T*S,f\rangle=\langle T(u),\langle S(s),f(u+s)\rangle.\] for all $f\in {\mathcal D}$. Show that $T*S\in{\mathcal D}'$. \end{question} \begin{question}\label{start bump} Consider the function $E:{\mathbb R}\rightarrow{\mathbb R}$ defined by \begin{alignat*}{2} E(0)&=0\\ E(x)&=\exp(-1/x^{2})&&\qquad\text{otherwise}. \end{alignat*} (i) Prove by induction, using the standard rules of differentiation, that $E$ is infinitely differentiable at all points $x\neq 0$ and that, at these points, \[E^{(n)}(x)=P_{n}(1/x)\exp(-1/x^{2})\] where $P_{n}$ is a polynomial which need not be found explicitly. (ii) Explain why $x^{-1}P_{n}(1/x)\exp(-1/x^{2})\rightarrow 0$ as $x\rightarrow 0$. (iii) Show by induction, using the definition of differentiation, that $E$ is infinitely differentiable at $0$ with $E^{(n)}(0)=0$ for all $n$. [Be careful to get this part of the argument right.] (iv) Show that \[E(x)=\sum_{j=0}^{\infty}\frac{E^{(j)}(0)}{j!}x^{j}\] if and only if $x=0$. (The reader may prefer to say that `The Taylor expansion of $E$ is only valid at $0$'.) (v) If you know some version of Taylor's theorem examine why it does not apply to $E$. \end{question} \begin{question}\label{end bump} The hard work for this question was done in Exercise~\ref{start bump}. (i) Let $F:{\mathbb R}\rightarrow{\mathbb R}$ be defined by $F(x)=0$ for $x<0$, $F(x)=E(x)$ for $x\geq 0$ where $E$ is the function defined in Exercise~\ref{start bump}. Show that $F$ is infinitely differentiable. (ii) Sketch the functions $f_{1},\,f_{2}:{\mathbb R}\rightarrow{\mathbb R}$ given by $f_{1}(x)=F(1-x)F(x)$ and $f_{2}(x)=\int_{0}^{x}f_{1}(t)\,dt$. (iii) Show that given $a<\alpha<\beta 0$ for $x\in(a,b)$ and $f(x)=0$ for all $x\notin [a,b]$. \end{question} \begin{question}\label{support} (This requires elementary topology, in particular knowledge of compactness and/or the Heine--Borel theorem.) (i) Let $T\in{\mathcal D}'$. We say that an open interval $(a,b)\in A$ if we can find an $\eta>0$ such that, if $f\in{\mathcal D}$ and $f(x)=0$ whenever $x\notin(a-\eta,b+\eta)$ then $\langle T,f\rangle=0$. Let $U=\bigcup_{(a,b)\in A}(a,b)$ and $\supp T={\mathbb T}\setminus U$. Explain why $\supp T$ is closed. Show, by using compactness and an argument along the lines of our proof of Lemma~\ref{start support}, that if $K$ is closed set with $K\cap\supp T=\emptyset$, $f\in{\mathcal D}$ and $f(x)=0$ for all $x\notin K$, then $\langle T,f\rangle=0$. (ii) We continue with the notation of~(i). Suppose $L$ is a closed set with the property that, if $K$ is closed set with $K\cap L=\emptyset$, $f\in{\mathcal D}$ and $f(x)=0$ for all $x\notin K$, then $\langle T,f\rangle=0$. Show that $L\supseteq \supp T$. (iii) If $S,\,T\in{\mathcal D}'$ show that \[\supp(T+S)\subseteq \supp T \cup \supp S.\] (iv) If $T\in{\mathcal D}'$ show that \[\supp T'\subseteq \supp T.\] (v) If $f\in C({\mathbb T})$ show that $\supp T_{f}$ (or, more briefly, $\supp f$ is the closure of $\{x\,:\,f(x)\neq 0\}$. If $f\in{\mathcal D}$ and $T\in{\mathcal D}'$ show that \[\supp fT\subseteq \supp T\cap \supp f.\] \end{question} \begin{question} (Only if you know about metric spaces.) (i) Show that, if we set \[d(f,g)=\sum_{r=0}^{\infty} \frac{2^{-r}\|f^{(r)}-g^{(r)}\|_{\infty}}{1+\|f^{(r)}-g^{(r)}\|_{\infty}},\] then $({\mathcal D},d)$ is a metric space. (ii) Show that $f_{n}\arrowD f$ if and only if $d(f_{n},f)\rightarrow 0$. (iii) (Only if you know what this means.) Show that $({\mathcal D},d)$ is complete. (iv) Find a metric $\rho$ on ${\mathcal S}$ such that $f_{n}\arrowS f$ if and only if $\rho(f_{n},f)\rightarrow 0$ \end{question} \begin{question} Show that the following equality holds in the space of tempered distributions \[2\pi\sum_{n=-\infty}^{\infty}\delta_{2\pi n} =\sum_{m=-\infty}^{\infty}e_{m}\] where $\delta_{2\pi n}$ is the delta function at $2\pi n$ and $e_{n}$ is the exponential function given by $e_{n}(t)=\exp(int)$. What formula results if we take the Fourier transform of both sides? \end{question} \end{document}