Topics In Analysis

• $3 . \mathrm{I} . 2 \mathrm{~F} \quad$

  Topics in Analysis

  Part II, 2005

  Let $-1 \leqslant x_{1}<x_{2}<\ldots<x_{n} \leqslant 1$ and let $a_{1}, a_{2}, \ldots, a_{n}$ be real numbers such that

  $$\int_{-1}^{1} p(t) d t=\sum_{i=1}^{n} a_{i} p\left(x_{i}\right)$$

  for every polynomial $p$ of degree less than $2 n$. Prove the following three facts.

  (i) $a_{i}>0$ for every $i$.

  (ii) $\sum_{i=1}^{n} a_{i}=2$.

  (iii) The numbers $x_{1}, x_{2}, \ldots, x_{n}$ are the roots of the Legendre polynomial of degree $n$.

  [You may assume standard orthogonality properties of the Legendre polynomials.]

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• 4.I.2F

  Topics in Analysis

  Part II, 2005

  (i) Let $D \subset \mathbb{C}$ be a domain, let $f: D \rightarrow \mathbb{C}$ be an analytic function and let $z_{0} \in D$. What does Taylor's theorem say about $z_{0}, f$ and $D$ ?

  (ii) Let $K$ be the square consisting of all complex numbers $z$ such that

  $$-1 \leqslant \operatorname{Re}(z) \leqslant 1 \text { and }-1 \leqslant \operatorname{Im}(z) \leqslant 1,$$

  and let $w$ be a complex number not belonging to $K$. Prove that the function $f(z)=$ $(z-w)^{-1}$ can be uniformly approximated on $K$ by polynomials.

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• 1.I.2F

  Topics in Analysis

  Part II, 2005

  Prove that $\cosh (1 / 2)$ is irrational.

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• 1.II.11F

  Topics in Analysis

  Part II, 2005

  State and prove a discrete form of Brouwer's theorem, concerning colourings of points in triangular grids. Use it to deduce that there is no continuous retraction from a disc to its boundary.

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• $2 . \mathrm{I} . 2 \mathrm{~F}$

  Topics in Analysis

  Part II, 2005

  (i) Let $\alpha$ be an algebraic number and let $p$ and $q$ be integers with $q \neq 0$. What does Liouville's theorem say about $\alpha$ and $p / q$ ?

  (ii) Let $p$ and $q$ be integers with $q \neq 0$. Prove that

  $$\left|\sqrt{2}-\frac{p}{q}\right| \geqslant \frac{1}{4 q^{2}} .$$

  [In (ii), you may not use Liouville's theorem unless you prove it.]

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• 2.II.11F

  Topics in Analysis

  Part II, 2005

  (i) State the Baire category theorem. Deduce from it a statement about nowhere dense sets.

  (ii) Let $X$ be the set of all real numbers with decimal expansions consisting of the digits 4 and 5 only. Prove that there is a real number $t$ that cannot be written in the form $x+y$ with $x \in X$ and $y$ rational.

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• 3.I.2G

  Topics in Analysis

  Part II, 2006

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be positive integers and, for each $n$, let

  $$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+} \cdot \ddots+\frac{1}{a_{n}}}$$

  with $\left(p_{n}, q_{n}\right)=1$.

  Obtain an expression for the matrix $\left(\begin{array}{cc}p_{n} & p_{n-1} \\ q_{n} & q_{n-1}\end{array}\right)$ and use it to show that $p_{n} q_{n-1}-q_{n} p_{n-1}=(-1)^{n+1} .$

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• 4.I.2G

  Topics in Analysis

  Part II, 2006

  (a) State the Baire category theorem, in its closed-sets version.

  (b) For every $n \in \mathbb{N}$ let $f_{n}$ be a continuous function from $\mathbb{R}$ to $\mathbb{R}$, and let $g(x)=1$ when $x$ is rational and 0 otherwise. For each $N \in \mathbb{N}$, let

  $$F_{N}=\left\{x \in \mathbb{R}: \forall n \geqslant N \quad f_{n}(x) \leqslant \frac{1}{3} \quad \text { or } \quad f_{n}(x) \geqslant \frac{2}{3}\right\} .$$

  By applying the Baire category theorem, prove that the functions $f_{n}$ cannot converge pointwise to $g$. (That is, it is not the case that $f_{n}(x) \rightarrow g(x)$ for every $x \in \mathbb{R}$.)

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• 1.I.2G

  Topics in Analysis

  Part II, 2006

  State Brouwer's fixed-point theorem, and also an equivalent version of the theorem that concerns retractions of the disc. Prove that these two versions are equivalent.

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• 1.II.11G

  Topics in Analysis

  Part II, 2006

  Let $\mathbb{T}=\{z:|z|=1\}$ be the unit circle in $\mathbb{C}$, and let $\phi: \mathbb{T} \rightarrow \mathbb{C}$ be a continuous function that never takes the value 0 . Define the degree (or winding number) of $\phi$ about 0 . [You need not prove that the degree is well-defined.]

  Denote the degree of $\phi$ about 0 by $w(\phi)$. Prove the following facts.

  (i) If $\phi_{1}$ and $\phi_{2}$ are two functions with the properties of $\phi$ above, then $w\left(\phi_{1} \cdot \phi_{2}\right)=$ $w\left(\phi_{1}\right)+w\left(\phi_{2}\right)$

  (ii) If $\psi$ is any continuous function such that $|\psi(z)|<|\phi(z)|$ for every $z \in \mathbb{T}$, then $w(\phi+\psi)=w(\phi)$.

  Using these facts, calculate the degree $w(\phi)$ when $\phi$ is given by the formula $\phi(z)=$ $(3 z-2)(z-3)(2 z+1)+1 .$

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• 2.I.2G

  Topics in Analysis

  Part II, 2006

  (a) State Chebyshev's equal ripple criterion.

  (b) Let $f:[-1,1] \rightarrow \mathbb{R}$ be defined by

  $$f(x)=\cos 4 \pi x$$

  and let $g$ be a polynomial of degree 7 . Prove that there exists an $x \in[-1,1]$ such that $|f(x)-g(x)| \geqslant 1$.

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• 2.II.11G

  Topics in Analysis

  Part II, 2006

  (a) Let $K$ be a closed subset of the unit disc in $\mathbb{C}$. Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a rational function with all its poles of modulus strictly greater than 1 . Explain why $f$ can be approximated uniformly on $K$ by polynomials.

  [Standard results from complex analysis may be assumed.]

  (b) With $K$ as above, define $\Lambda$ to be the set of all $\lambda \in \mathbb{C} \backslash K$ such that the function $(z-\lambda)^{-1}$ can be uniformly approximated on $K$ by polynomials. If $\lambda \in \Lambda$, prove that there is some $\delta>0$ such that $\mu \in \Lambda$ whenever $|\lambda-\mu|<\delta$.

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• $3 . \mathrm{I} . 2 \mathrm{~F}$

  Topics in Analysis

  Part II, 2007

  State a version of Runge's theorem and use it to prove the following theorem:

  Let $D=\{z \in \mathbb{C}:|z|<1\}$ and define $f: D \rightarrow \mathbb{C}$ by the condition

  $$f\left(r e^{i \theta}\right)=r^{3 / 2} e^{3 i \theta / 2}$$

  for all $0 \leqslant r<1$ and all $0 \leqslant \theta<2 \pi$. (We take $r^{1 / 2}$ to be the positive square root.) Then there exists a sequence of analytic functions $f_{n}: D \rightarrow \mathbb{C}$ such that $f_{n}(z) \rightarrow f(z)$ for each $z \in D$ as $n \rightarrow \infty$.

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• 4.I.2F

  Topics in Analysis

  Part II, 2007

  State Brouwer's fixed point theorem for a triangle in two dimensions.

  Let $A=\left(a_{i j}\right)$ be a $3 \times 3$ matrix with real positive entries and such that all its columns are non-zero vectors. Show that $A$ has an eigenvector with positive entries.

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• 1.I.2F

  Topics in Analysis

  Part II, 2007

  Let $n$ be an integer with $n \geqslant 1$. Are the following statements true or false? Give proofs.

  (i) There exists a real polynomial $T_{n}$ of degree $n$ such that

  $$T_{n}(\cos t)=\cos n t$$

  for all real $t$.

  (ii) There exists a real polynomial $R_{n}$ of degree $n$ such that

  $$R_{n}(\cosh t)=\cosh n t$$

  for all real $t$.

  (iii) There exists a real polynomial $S_{n}$ of degree $n$ such that

  $$S_{n}(\cos t)=\sin n t$$

  for all real $t$.

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• 2.II.12F

  Topics in Analysis

  Part II, 2007

  (i) Suppose that $f:[0,1] \rightarrow \mathbb{R}$ is continuous. Prove the theorem of Bernstein which states that, if we write

  $$f_{m}(t)=\sum_{r=0}^{m}\left(\begin{array}{c} m \\ r \end{array}\right) f(r / m) t^{r}(1-t)^{m-r}$$

  for $0 \leqslant t \leqslant 1$, then $f_{m} \rightarrow f$ uniformly as $m \rightarrow \infty$

  (ii) Let $n \geqslant 1, a_{1, n}, a_{2, n}, \ldots, a_{n, n} \in \mathbb{R}$ and let $x_{1, n}, x_{2, n}, \ldots, x_{n, n}$ be distinct points in $[0,1]$. We write

  $$I_{n}(g)=\sum_{j=1}^{n} a_{j, n} g\left(x_{j, n}\right)$$

  for every continuous function $g:[0,1] \rightarrow \mathbb{R}$. Show that, if

  $$I_{n}(P)=\int_{0}^{1} P(t) d t$$

  for all polynomials $P$ of degree $2 n-1$ or less, then $a_{j, n} \geqslant 0$ for all $1 \leqslant j \leqslant n$ and $\sum_{j=1}^{n} a_{j, n}=1 .$

  (iii) If $I_{n}$ satisfies the conditions set out in (ii), show that

  $$I_{n}(f) \rightarrow \int_{0}^{1} f(t) d t$$

  as $n \rightarrow \infty$ whenever $f:[0,1] \rightarrow \mathbb{R}$ is continuous.

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• $2 . \mathrm{I} . 2 \mathrm{~F}$

  Topics in Analysis

  Part II, 2007

  Write

  $$P^{+}=\left\{(x, y) \in \mathbb{R}^{2}: x, y>0\right\} .$$

  Suppose that $K$ is a convex, compact subset of $\mathbb{R}^{2}$ with $K \cap P^{+} \neq \emptyset$. Show that there is a unique point $\left(x_{0}, y_{0}\right) \in K \cap P^{+}$such that

  $$x y \leqslant x_{0} y_{0}$$

  for all $(x, y) \in K \cap P^{+}$.

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• 3.II.12F

  Topics in Analysis

  Part II, 2007

  (i) State and prove Liouville's theorem on approximation of algebraic numbers by rationals.

  (ii) Consider the continued fraction

  $$x=\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\ldots}}}}$$

  where the $a_{j}$ are strictly positive integers. You may assume the following algebraic facts about the $n$th convergent $p_{n} / q_{n}$.

  $$p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n}, \quad q_{n}=a_{n} q_{n-1}+q_{n-2} .$$

  Show that

  $$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{q_{n} q_{n+1}}$$

  Give explicit values for $a_{n}$ so that $x$ is transcendental and prove that you have done SO.

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• $3 . \mathrm{I} . 2 \mathrm{~F} \quad$

  Topics in Analysis

  Part II, 2008

  (a) State the Baire category theorem in its closed sets version.

  (b) Let $f_{n}: \mathbf{R} \rightarrow \mathbf{R}$ be a continuous function for each $n=1,2,3, \ldots$ and suppose that there is a function $f: \mathbf{R} \rightarrow \mathbf{R}$ such that $f_{n}(x) \rightarrow f(x)$ for each $x \in \mathbf{R}$. Prove that for each $\epsilon>0$, there exists an integer $N_{0}$ and a non-empty open interval $I \subset \mathbf{R}$ such that $\left|f_{n}(x)-f(x)\right| \leqslant \epsilon$ for all $n \geqslant N_{0}$ and $x \in I$.

  [Hint: consider, for $N=1,2,3, \ldots$, the sets

  $$\left.Q_{N}=\left\{x \in \mathbf{R}:\left|f_{n}(x)-f_{m}(x)\right| \leqslant \epsilon: \forall n, m \geqslant N\right\} .\right]$$

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• 4.I.2F

  Topics in Analysis

  Part II, 2008

  (a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

  (b) Suppose $f$ is analytic on

  $$\Omega=\{z \in \mathbf{C}:|z|<1\} \backslash\{z \in \mathbf{C}: \operatorname{Im}(z)=0, \operatorname{Re}(z) \leqslant 0\} .$$

  Prove that there exists a sequence of polynomials which converges to $f$ uniformly on compact subsets of $\Omega$.

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• 1.I.2F

  Topics in Analysis

  Part II, 2008

  Let $P_{0}, P_{1}, P_{2}, \ldots$ be non-zero orthogonal polynomials on an interval $[a, b]$ such that the degree of $P_{j}$ is equal to $j$ for every $j=0,1,2, \ldots$, where the orthogonality is with respect to the inner product $<f, g>=\int_{a}^{b} f g$. If $f$ is any continuous function on $[a, b]$ orthogonal to $P_{0}, P_{1}, \ldots, P_{n-1}$ and not identically zero, prove that $f$ must have at least $n$ distinct zeros in $(a, b)$.

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• 2.II.11F

  Topics in Analysis

  Part II, 2008

  Let $L: C([0,1]) \rightarrow C([0,1])$ be an operator satisfying the conditions

  (i) $L f \geqslant 0$ for any $f \in C([0,1])$ with $f \geqslant 0$,

  (ii) $L(a f+b g)=a L f+b L g$ for any $f, g \in C([0,1])$ and $a, b \in \mathbf{R}$ and

  (iii) $Z_{f} \subseteq Z_{L f}$ for any $f \in C([0,1])$, where $Z_{f}$ denotes the set of zeros of $f$.

  Prove that there exists a function $h \in C([0,1])$ with $h \geqslant 0$ such that $L f=h f$ for every $f \in C([0,1])$.

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• $2 . \mathrm{I} . 2 \mathrm{~F} \quad$

  Topics in Analysis

  Part II, 2008

  (a) State Brouwer's fixed point theorem in the plane and prove that the statement is equivalent to non-existence of a continuous retraction of the closed disk $D$ to its boundary $\partial D$.

  (b) Use Brouwer's fixed point theorem to prove that there is a complex number $z$ in the closed unit disc such that $z^{6}-z^{5}+2 z^{2}+6 z+1=0$.

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• $3 . \mathrm{II} . 12 \mathrm{~F}$

  Topics in Analysis

  Part II, 2008

  (a) State Liouville's theorem on approximation of algebraic numbers by rationals.

  (b) Consider the continued fraction expression

  $$x=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

  in which the coefficients $a_{n}$ are all positive integers forming an unbounded set. Let $\frac{p_{n}}{q_{n}}$ be the $n$th convergent. Prove that

  $$\left|x-\frac{p_{n}}{q_{n}}\right| \leqslant \frac{1}{q_{n} q_{n+1}}$$

  and use this inequality together with Liouville's theorem to deduce that $x^{2}$ is irrational.

  [ You may assume without proof that, for $n=1,2,3, \ldots$,

  $$\left.\left(\begin{array}{ll} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{array}\right)=\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)\left(\begin{array}{cc} a_{n+1} & 1 \\ 1 & 0 \end{array}\right) .\right]$$

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• Paper 4, Section I, $2 F$

  Topics in Analysis

  Part II, 2009

  State Liouville's theorem on approximation of algebraic numbers by rationals, and use it to prove that the number

  $$\sum_{n=0}^{\infty} \frac{1}{10^{n !}}$$

  is transcendental.

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• Paper 3, Section I, $2 F$

  Topics in Analysis

  Part II, 2009

  (a) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous, prove that there exists a sequence of polynomials $P_{n}$ such that $P_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.

  (b) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous and bounded, prove that there exists a sequence of polynomials $Q_{n}$ such that $Q_{n}$ are uniformly bounded on $(0,1)$ and $Q_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.

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• Paper 2, Section I, $2 F$

  Topics in Analysis

  Part II, 2009

  (a) State Chebychev's Equal Ripple Criterion.

  (b) Let $n$ be a positive integer, $a_{0}, a_{1}, \ldots, a_{n-1} \in \mathbb{R}$ and

  $$p(x)=x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0} .$$

  Use Chebychev's Equal Ripple Criterion to prove that

  $$\sup _{x \in[-1,1]}|p(x)| \geqslant 2^{1-n}$$

  [You may use without proof that there is a polynomial $T_{n}(x)$ in $x$ of degree $n$, with the coefficient of $x^{n}$ equal to $2^{n-1}$, such that $T_{n}(\cos \theta)=\cos n \theta$ for all $\theta \in \mathbb{R}$.]

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• Paper 1, Section I, $2 F$

  Topics in Analysis

  Part II, 2009

  (i) Let $n \geqslant 1$ and let $x_{1}, \ldots, x_{n}$ be distinct points in $[-1,1]$. Show that there exist numbers $A_{1}, \ldots, A_{n}$ such that

  $$\int_{-1}^{1} P(x) d x=\sum_{j=1}^{n} A_{j} P\left(x_{j}\right)$$

  for every polynomial $P$ of degree $\leqslant n-1$.

  (ii) Explain, without proof, how one can choose the points $x_{1}, \ldots, x_{n}$ and the numbers $A_{1}, \ldots, A_{n}$ such that $(*)$ holds for all polynomials $P$ of degree $\leqslant 2 n-1$.

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• Paper 2, Section II, F

  Topics in Analysis

  Part II, 2009

  (a) State Brouwer's fixed point theorem in the plane.

  (b) Let $\mathbf{a}, \mathbf{b}, \mathbf{c}$ be unit vectors in $\mathbb{R}^{2}$ making $120^{\circ}$ angles with one another. Let $T$ be the triangle with vertices given by the points $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$ and let $I, J, K$ be the three sides of $T$. Prove that the following two statements are equivalent:

  (1) There exists no continuous function $f: T \rightarrow \partial T$ with $f(I) \subseteq I, f(J) \subseteq J$ and $f(K) \subseteq K$.

  (2) If $A, B, C$ are closed subsets of $\mathbb{R}^{2}$ such that $T \subseteq A \cup B \cup C, I \subseteq A, J \subseteq B$ and $K \subseteq C$, then $A \cap B \cap C \neq \emptyset$.

  (c) Let $f, g: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous positive functions. Show that the system of equations

  $$\begin{aligned} &\left(1-x^{2}\right) f^{2}(x, y)-x^{2} g^{2}(x, y)=0 \\ &\left(1-y^{2}\right) g^{2}(x, y)-y^{2} f^{2}(x, y)=0 \end{aligned}$$

  has four distinct solutions on the unit circle $\mathbb{S}^{1}=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$.

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• Paper 3, Section II, F

  Topics in Analysis

  Part II, 2009

  (a) State Runge's theorem on uniform approximation of analytic functions by polynomials.

  (b) Let $\Omega$ be an unbounded, connected, proper open subset of $\mathbb{C}$. For any given compact set $K \subset \mathbb{C} \backslash \Omega$ and any $\zeta \in \Omega$, show that there exists a sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.

  (c) Give an example, with justification, of a connected open subset $\Omega$ of $\mathbb{C}$, a compact subset $K$ of $\mathbb{C} \backslash \Omega$ and a point $\zeta \in \Omega$ such that there is no sequence of complex polynomials converging uniformly on $K$ to the function $f(z)=(z-\zeta)^{-1}$.

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• Paper 1, Section I, F

  Topics in Analysis

  Part II, 2010

  Let $(X, d)$ be a non-empty complete metric space with no isolated points, $G$ an open dense subset of $X$ and $E$ a countable dense subset of $X$.

  (i) Stating clearly any standard theorem you use, prove that $G \backslash E$ is a dense subset of $X$.

  (ii) If $G$ is only assumed to be uncountable and dense in $X$, does it still follow that $G \backslash E$ is dense in $X$ ? Justify your answer.

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• Paper 2, Section I, F

  Topics in Analysis

  Part II, 2010

  (a) State the Weierstrass approximation theorem concerning continuous real functions on the closed interval $[0,1]$.

  (b) Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous.

  (i) If $\int_{0}^{1} f(x) x^{n} d x=0$ for each $n=0,1,2, \ldots$, prove that $f$ is the zero function.

  (ii) If we only assume that $\int_{0}^{1} f(x) x^{2 n} d x=0$ for each $n=0,1,2, \ldots$, prove that it still follows that $f$ is the zero function.

  [If you use the Stone-Weierstrass theorem, you must prove it.]

  (iii) If we only assume that $\int_{0}^{1} f(x) x^{2 n+1} d x=0$ for each $n=0,1,2, \ldots$, does it still follow that $f$ is the zero function? Justify your answer.

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• Paper 3, Section I, F

  Topics in Analysis

  Part II, 2010

  Let $A=\{z \in \mathbb{C}: 1 / 2 \leqslant|z| \leqslant 2\}$ and suppose that $f$ is complex analytic on an open subset containing $A$.

  (i) Give an example, with justification, to show that there need not exist a sequence of complex polynomials converging to $f$ uniformly on $A$.

  (ii) Let $R \subset \mathbb{C}$ be the positive real axis and $B=A \backslash R$. Prove that there exists a sequence of complex polynomials $p_{1}, p_{2}, p_{3}, \ldots$ such that $p_{j} \rightarrow f$ uniformly on each compact subset of $B$.

  (iii) Let $p_{1}, p_{2}, p_{3}, \ldots$ be the sequence of polynomials in (ii). If this sequence converges uniformly on $A$, show that $\int_{C} f(z) d z=0$, where $C=\{z \in \mathbb{C}:|z|=1\}$.

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• Paper 4, Section I, $2 F$

  Topics in Analysis

  Part II, 2010

  Find explicitly a polynomial $p$ of degree $\leqslant 3$ such that

  $$\sup _{x \in[-1,1]}\left|x^{4}-p(x)\right| \leqslant \sup _{x \in[-1,1]}\left|x^{4}-q(x)\right|$$

  for every polynomial $q$ of degree $\leqslant 3$. Justify your answer.

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• Paper 2, Section II, 11F

  Topics in Analysis

  Part II, 2010

  Let

  $$B_{r}(0)=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}<r^{2}\right\},$$

  $B=B_{1}(0)$, and

  $$C=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$$

  Let $D=B \cup C$.

  (i) State the Brouwer fixed point theorem on the plane.

  (ii) Show that the Brouwer fixed point theorem on the plane is equivalent to the nonexistence of a continuous map $F: D \rightarrow C$ such that $F(p)=p$ for each $p \in C$.

  (iii) Let $G: D \rightarrow \mathbb{R}^{2}$ be continuous, $0<\epsilon<1$ and suppose that

  $$|p-G(p)|<\epsilon$$

  for each $p \in C$. Using the Brouwer fixed point theorem or otherwise, prove that

  $$B_{1-\epsilon}(0) \subseteq G(B)$$

  [Hint: argue by contradiction.]

  (iv) Let $q \in B$. Does there exist a continuous map $H: D \rightarrow \mathbb{R}^{2} \backslash\{q\}$ such that $H(p)=p$ for each $p \in C$ ? Justify your answer.

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• Paper 3, Section II, F

  Topics in Analysis

  Part II, 2010

  (i) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map with $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin.

  (ii) For $j=0,1$, let $\gamma_{j}:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma_{j}(0)=\gamma_{j}(1)$. Make the following statement precise, and prove it: if $\gamma_{0}$ can be continuously deformed into $\gamma_{1}$ through a family of continuous curves missing the origin, then $w\left(\gamma_{0} ; 0\right)=w\left(\gamma_{1} ; 0\right)$.

  [You may use without proof the following fact: if $\gamma, \delta:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ are continuous with $\gamma(0)=\gamma(1), \delta(0)=\delta(1)$ and if $|\gamma(t)|<|\delta(t)|$ for each $t \in[0,1]$, then $w(\gamma+\delta ; 0)=w(\delta ; 0)$.]

  (iii) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be continuous with $\gamma(0)=\gamma(1)$. If $\gamma(t)$ is not equal to a negative real number for each $t \in[0,1]$, prove that $w(\gamma ; 0)=0$.

  (iv) Let $D=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and $C=\{z \in \mathbb{C}:|z|=1\}$. If $g: D \rightarrow C$ is continuous, prove that for each non-zero integer $n$, there is at least one point $z \in C$ such that $z^{n}+g(z)=0$.

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• Paper 1, Section I, F

  Topics in Analysis

  Part II, 2011

  (i) State the Baire Category Theorem for metric spaces in its closed sets version.

  (ii) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a complex analytic function which is not a polynomial. Prove that there exists a point $z_{0} \in \mathbb{C}$ such that each coefficient of the Taylor series of $f$ at $z_{0}$ is non-zero.

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• Paper 2, Section I, F

  Topics in Analysis

  Part II, 2011

  (i) Let $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ be any set of $n$ distinct numbers. Show that there exist numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the formula

  $$\int_{-1}^{1} p(x) d x=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

  is valid for every polynomial $p$ of degree $\leqslant n-1$.

  (ii) For $n=0,1,2, \ldots$, let $p_{n}$ be the Legendre polynomial, over $[-1,1]$, of degree $n$. Suppose that $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ are the roots of $p_{n}$, and $A_{1}, A_{2}, \ldots, A_{n}$ are the numbers corresponding to $x_{1}, x_{2}, \ldots, x_{n}$ as in (i).

  [You may assume without proof that for $n \geqslant 1, p_{n}$ has $n$ distinct roots in $[-1,1] .$ ]

  Prove that the integration formula in (i) is now valid for any polynomial $p$ of degree $\leqslant 2 n-1$.

  (iii) Is it possible to choose $n$ distinct points $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ and corresponding numbers $A_{1}, A_{2}, \ldots, A_{n}$ such that the integration formula in (i) is valid for any polynomial $p$ of degree $\leqslant 2 n$ ? Justify your answer.

  comment

• Paper 3, Section I, F

  Topics in Analysis

  Part II, 2011

  Let $\Gamma=\{z \in \mathbb{C}: z \neq 1,|\operatorname{Re}(z)|+|\operatorname{Im}(z)|=1\}$.

  (i) Prove that, for any $\zeta \in \mathbb{C}$ with $|\operatorname{Re}(\zeta)|+|\operatorname{Im}(\zeta)|>1$ and any $\epsilon>0$, there exists a complex polynomial $p$ such that

  $$\sup _{z \in \Gamma}\left|p(z)-(z-\zeta)^{-1}\right|<\epsilon$$

  (ii) Does there exist a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow(z-1)^{-1}$ for every $z \in \Gamma ?$ Justify your answer.

  comment

• Paper 4, Section I, $2 F$

  Topics in Analysis

  Part II, 2011

  (a) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin. State precisely a theorem about homotopy invariance of the winding number.

  (b) Let $B=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and let $f: B \rightarrow \mathbb{C}$ be a continuous map satisfying

  $$|f(z)-z| \leqslant 1$$

  for each $z \in \partial B$.

  (i) For $0 \leqslant t \leqslant 1$, let $\gamma(t)=f\left(e^{2 \pi i t}\right)$. If $\gamma(t) \neq 0$ for each $t \in[0,1]$, prove that $w(\gamma ; 0)=1$.

  [Hint: Consider a suitable homotopy between $\gamma$ and the map $\gamma_{1}(t)=e^{2 \pi i t}$, $0 \leqslant t \leqslant 1 .]$

  (ii) Deduce that $f(z)=0$ for some $z \in B$.

  comment

• Paper 2, Section II, F

  Topics in Analysis

  Part II, 2011

  Let $C[0,1]$ be the space of real continuous functions on the interval $[0,1]$. A mapping $L: C[0,1] \rightarrow C[0,1]$ is said to be positive if $L(f) \geqslant 0$ for each $f \in C[0,1]$ with $f \geqslant 0$, and linear if $L(a f+b g)=a L(f)+b L(g)$ for all functions $f, g \in C[0,1]$ and constants $a, b \in \mathbb{R}$.

  (i) Let $L_{n}: C[0,1] \rightarrow C[0,1]$ be a sequence of positive, linear mappings such that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for the three functions $f(x)=1, x, x^{2}$. Prove that $L_{n}(f) \rightarrow f$ uniformly on $[0,1]$ for every $f \in C[0,1]$.

  (ii) Define $B_{n}: C[0,1] \rightarrow C[0,1]$ by

  $$B_{n}(f)(x)=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) f\left(\frac{k}{n}\right) x^{k}(1-x)^{n-k}$$

  where $\left(\begin{array}{l}n \\ k\end{array}\right)=\frac{n !}{k !(n-k) !}$. Using the result of part (i), or otherwise, prove that $B_{n}(f) \rightarrow f$ uniformly on $[0,1]$.

  (iii) Let $f \in C[0,1]$ and suppose that

  $$\int_{0}^{1} f(x) x^{4 n} d x=0$$

  for each $n=0,1, \ldots$ Prove that $f$ must be the zero function.

  [You should not use the Stone-Weierstrass theorem without proof.]

  comment

• Paper 3, Section II, F

  Topics in Analysis

  Part II, 2011

  Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous and let $n$ be a positive integer. For $g:[0,1] \rightarrow \mathbb{R}$ a continuous function, write $\|f-g\|_{L^{\infty}}=\sup _{x \in[0,1]}|f(x)-g(x)|$.

  (i) Let $p$ be a polynomial of degree at most $n$ with the property that there are $(n+2)$ distinct points $x_{1}, x_{2}, \ldots, x_{n+2} \in[0,1]$ with $x_{1}<x_{2}<\ldots<x_{n+2}$ such that

  $$f\left(x_{j}\right)-p\left(x_{j}\right)=(-1)^{j}\|f-p\|_{L^{\infty}}$$

  for each $j=1,2, \ldots, n+2$. Prove that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  (ii) Prove that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}} \leqslant\|f-q\|_{L^{\infty}}$$

  for every polynomial $q$ of degree at most $n$.

  [If you deduce this from a more general result about abstract normed spaces, you must prove that result.]

  (iii) Let $Y=\left\{y_{1}, y_{2}, \ldots, y_{n+2}\right\}$ be any set of $(n+2)$ distinct points in $[0,1]$.

  (a) For $j=1,2, \ldots, n+2$, let

  $$r_{j}(x)=\prod_{k=1, k \neq j}^{n+2} \frac{x-y_{k}}{y_{j}-y_{k}}$$

  $t(x)=\sum_{j=1}^{n+2} f\left(y_{j}\right) r_{j}(x)$ and $r(x)=\sum_{j=1}^{n+2}(-1)^{j} r_{j}(x)$. Explain why there is a unique number $\lambda \in \mathbb{R}$ such that the degree of the polynomial $t-\lambda r$ is at $\operatorname{most} n$.

  (b) Let $\|f-g\|_{L^{\infty}(Y)}=\sup _{x \in Y}|f(x)-g(x)|$. Deduce from part (a) that there exists a polynomial $p$ of degree at most $n$ such that

  $$\|f-p\|_{L^{\infty}(Y)} \leqslant\|f-q\|_{L^{\infty}(Y)}$$

  for every polynomial $q$ of degree at most $n$.

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• Paper 4, Section I, $2 F$

  Topics in Analysis

  Part II, 2012

  Let $A_{1}, A_{2}, \ldots, A_{n}$ be real numbers and suppose that $x_{1}, x_{2}, \ldots, x_{n} \in[-1,1]$ are distinct. Suppose that the formula

  $$\int_{-1}^{1} p(x) d x=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

  is valid for every polynomial $p$ of degree $\leqslant 2 n-1$. Prove the following:

  (i) $A_{j}>0$ for each $j=1,2, \ldots, n$.

  (ii) $\sum_{j=1}^{n} A_{j}=2$.

  (iii) $x_{1}, x_{2}, \ldots, x_{n}$ are the roots of the Legendre polynomial of degree $n$.

  [You may assume standard orthogonality properties of the Legendre polynomials.]

  comment

• Paper 3, Section I, $2 F$

  Topics in Analysis

  Part II, 2012

  State and prove Liouville's theorem concerning approximation of algebraic numbers by rationals.

  comment

• Paper 2, Section I, $2 F$

  Topics in Analysis

  Part II, 2012

  (a) Let $\gamma:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\gamma(0)=\gamma(1)$. Define the winding number $w(\gamma ; 0)$ of $\gamma$ about the origin. State precisely a theorem about homotopy invariance of the winding number.

  (b) Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a continuous map such that $z^{-10} f(z)$ is bounded as $|z| \rightarrow \infty$. Prove that there exists a complex number $z_{0}$ such that

  $$f\left(z_{0}\right)=z_{0}^{11}$$

  comment

• Paper 1, Section I, F

  Topics in Analysis

  Part II, 2012

  State a version of the Baire category theorem for a complete metric space. Let $T$ be the set of real numbers $x$ with the property that, for each positive integer $n$, there exist integers $p$ and $q$ with $q \geqslant 2$ such that

  $$0<\left|x-\frac{p}{q}\right|<\frac{1}{q^{n}} .$$

  Is $T$ an open subset of $\mathbb{R}$ ? Is $T$ a dense subset of $\mathbb{R}$ ? Justify your answers.

  comment

• Paper 2, Section II, F

  Topics in Analysis

  Part II, 2012

  (a) State Runge's theorem about uniform approximability of analytic functions by complex polynomials.

  (b) Let $K$ be a compact subset of the complex plane.

  (i) Let $\Sigma$ be an unbounded, connected subset of $\mathbb{C} \backslash K$. Prove that for each $\zeta \in \Sigma$, the function $f(z)=(z-\zeta)^{-1}$ is uniformly approximable on $K$ by a sequence of complex polynomials.

  [You may not use Runge's theorem without proof.]

  (ii) Let $\Gamma$ be a bounded, connected component of $\mathbb{C} \backslash K$. Prove that there is no point $\zeta \in \Gamma$ such that the function $f(z)=(z-\zeta)^{-1}$ is uniformly approximable on $K$ by a sequence of complex polynomials.

  comment

• Paper 3, Section II, F

  Topics in Analysis

  Part II, 2012

  State Brouwer's fixed point theorem on the plane, and also an equivalent version of it concerning continuous retractions. Prove the equivalence of the two statements.

  Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous map with the property that $|f(x)| \leqslant 1$ whenever $|x|=1$. Show that $f$ has a fixed point. [Hint. Compose $f$ with the map that sends $x$ to the nearest point to $x$ inside the closed unit disc.]

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• Paper 4, Section I, F

  Topics in Analysis

  Part II, 2013

  State the Baire Category Theorem. A set $X \subseteq \mathbb{R}$ is said to be a $G_{\delta}$-set if it is the intersection of countably many open sets. Show that the set $\mathbb{Q}$ of rationals is not a $G_{\delta}$-set.

  [You may assume that the rationals are countable and that $\mathbb{R}$ is complete.]

  comment

• Paper 3, Section I, $2 F$

  Topics in Analysis

  Part II, 2013

  State Brouwer's fixed point theorem. Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function with the property that $|f(x)-x| \leqslant 1$ for all $x$. Show that $f$ is surjective.

  comment

• Paper 2, Section I, F

  Topics in Analysis

  Part II, 2013

  (i) Show that for every $\epsilon>0$ there is a polynomial $p: \mathbb{R} \rightarrow \mathbb{R}$ such that $\left|\frac{1}{x}-p(x)\right| \leqslant \epsilon$ for all $x \in \mathbb{R}$ satisfying $\frac{1}{2} \leqslant|x| \leqslant 2$.

  [You may assume standard results provided they are stated clearly.]

  (ii) Show that there is no polynomial $p: \mathbb{C} \rightarrow \mathbb{C}$ such that $\left|\frac{1}{z}-p(z)\right|<1$ for all $z \in \mathbb{C}$ satisfying $\frac{1}{2} \leqslant|z| \leqslant 2$.

  comment

• Paper 1, Section I, $2 F$

  Topics in Analysis

  Part II, 2013

  Show that $\sin (1)$ is irrational. [The angle is measured in radians.]

  comment

• Paper 2, Section II, F

  Topics in Analysis

  Part II, 2013

  (i) Let $n \geqslant 4$ be an integer. Show that

  $$1+\frac{1}{n+\frac{1}{1+\frac{1}{n+\ldots}}} \geqslant 1+\frac{1}{2 n} .$$

  (ii) Let us say that an irrational number $\alpha$ is badly approximable if there is some constant $c>0$ such that

  $$\left|\alpha-\frac{p}{q}\right| \geqslant \frac{c}{q^{2}}$$

  for all $q \geqslant 1$ and for all integers $p$. Show that if the integers $a_{n}$ in the continued fraction expansion $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ are bounded then $\alpha$ is badly approximable.

  Give, with proof, an example of an irrational number which is not badly approximable.

  [Standard facts about continued fractions may be used without proof provided they are stated clearly.]

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• Paper 3, Section II, F

  Topics in Analysis

  Part II, 2013

  Suppose that $x_{0}, x_{1}, \ldots, x_{n} \in[-1,1]$ are distinct points. Let $f$ be an infinitely differentiable real-valued function on an open interval containing $[-1,1]$. Let $p$ be the unique polynomial of degree at most $n$ such that $f\left(x_{r}\right)=p\left(x_{r}\right)$ for $r=0,1, \ldots, n$. Show that for each $x \in[-1,1]$ there is some $\xi \in(-1,1)$ such that

  $$f(x)-p(x)=\frac{f^{(n+1)}(\xi)}{(n+1) !}\left(x-x_{0}\right) \ldots\left(x-x_{n}\right)$$

  Now take $x_{r}=\cos \frac{2 r+1}{2 n+2} \pi$. Show that

  $$|f(x)-p(x)| \leqslant \frac{1}{2^{n}(n+1) !} \sup _{\xi \in[-1,1]}\left|f^{(n+1)}(\xi)\right|$$

  for all $x \in[-1,1]$. Deduce that there is a polynomial $p$ of degree at most $n$ such that

  $$\left|\frac{1}{3+x}-p(x)\right| \leqslant \frac{1}{4^{n+1}}$$

  for all $x \in[-1,1]$.

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• Paper 4, Section I, $2 G$

  Topics in Analysis

  Part II, 2014

  State Liouville's theorem on approximation of algebraic numbers by rationals.

  Prove that the number $\sum_{n=0}^{\infty} \frac{1}{2^{n^{n}}}$ is transcendental.

  comment

• Paper 3, Section I, G

  Topics in Analysis

  Part II, 2014

  State Runge's theorem about uniform approximation of holomorphic functions by polynomials.

  Let $\mathbb{R}_{+} \subset \mathbb{C}$ be the subset of non-negative real numbers and let

  $$\Delta=\{z \in \mathbb{C}:|z|<1\} .$$

  Prove that there is a sequence of complex polynomials which converges to the function $1 / z$ uniformly on each compact subset of $\Delta \backslash \mathbb{R}_{+}$.

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• Paper 2, Section I, G

  Topics in Analysis

  Part II, 2014

  State Chebyshev's equal ripple criterion.

  Let

  $$h(t)=\prod_{\ell=1}^{n}\left(t-\cos \frac{(2 \ell-1) \pi}{2 n}\right)$$

  Show that if $q(t)=\sum_{j=0}^{n} a_{j} t^{j}$ where $a_{0}, \ldots, a_{n}$ are real constants with $\left|a_{n}\right| \geqslant 1$, then

  $$\sup _{t \in[-1,1]}|h(t)| \leqslant \sup _{t \in[-1,1]}|q(t)|$$

  comment

• Paper 1, Section I, $2 \mathrm{G}$

  Topics in Analysis

  Part II, 2014

  (i) State Brouwer's fixed point theorem in the plane and an equivalent theorem concerning mapping a triangle $T$ to its boundary $\partial T$.

  (ii) Let $A$ be a $3 \times 3$ matrix with positive real entries. Use the theorems you stated in (i) to prove that $A$ has an eigenvector with positive entries.

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• Paper 2, Section II, G

  Topics in Analysis

  Part II, 2014

  Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a continuous map never taking the value 0 and satisfying $\gamma(0)=\gamma(1)$. Define the degree (or winding number) $w(\gamma ; 0)$ of $\gamma$ about 0 . Prove the following:

  (i) $w(1 / \gamma ; 0)=w\left(\gamma^{-} ; 0\right)$, where $\gamma^{-}(t)=\gamma(1-t)$.

  (ii) If $\sigma:[0,1] \rightarrow \mathbb{C}$ is continuous, $\sigma(0)=\sigma(1)$ and $|\sigma(t)|<|\gamma(t)|$ for each $0 \leqslant t \leqslant 1$, then $w(\gamma+\sigma ; 0)=w(\gamma ; 0)$.

  (iii) If $\gamma_{m}:[0,1] \rightarrow \mathbb{C}, m=1,2, \ldots$, are continuous maps with $\gamma_{m}(0)=\gamma_{m}(1)$, which converge to $\gamma$ uniformly on $[0,1]$ as $m \rightarrow \infty$, then $w\left(\gamma_{m} ; 0\right)=w(\gamma ; 0)$ for sufficiently large $m$.

  Let $\alpha:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ be a continuous map such that $\alpha(0)=\alpha(1)$ and $\left|\alpha(t)-e^{2 \pi i t}\right| \leqslant 1$ for each $t \in[0,1]$. Deduce from the results of (ii) and (iii) that $w(\alpha ; 0)=1$.

  [You may not use homotopy invariance of the winding number without proof.]

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• Paper 3, Section II, G

  Topics in Analysis

  Part II, 2014

  Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category Theorem. Show that any complete non-empty metric space without isolated points is uncountable.

  Let $A$ be the set of real numbers whose decimal expansion does not use the digit 6 . (A terminating decimal representation is used when it exists.) Show that there exists a real number which cannot be written as $a+q$ with $a \in A$ and $q \in \mathbb{Q}$.

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• Paper 4, Section I, $2 I$

  Topics in Analysis

  Part II, 2015

  Let $\mathcal{K}$ be the set of all non-empty compact subsets of $m$-dimensional Euclidean space $\mathbb{R}^{m}$. Define the Hausdorff metric on $\mathcal{K}$, and prove that it is a metric.

  Let $K_{1} \supseteq K_{2} \supseteq \ldots$ be a sequence in $\mathcal{K}$. Show that $K=\bigcap_{n=1}^{\infty} K_{n}$ is also in $\mathcal{K}$ and that $K_{n} \rightarrow K$ as $n \rightarrow \infty$ in the Hausdorff metric.

  comment

• Paper 3, Section I, $2 I$

  Topics in Analysis

  Part II, 2015

  Let $K$ be a compact subset of $\mathbb{C}$ with path-connected complement. If $w \notin K$ and $\epsilon>0$, show that there is a polynomial $P$ such that

  $$\left|P(z)-\frac{1}{w-z}\right| \leqslant \epsilon$$

  for all $z \in K$.

  comment

• Paper 2, Section I, I

  Topics in Analysis

  Part II, 2015

  Let $x_{1}, x_{2}, \ldots, x_{n}$ be the roots of the Legendre polynomial of degree $n$. Let $A_{1}$, $A_{2}, \ldots, A_{n}$ be chosen so that

  $$\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$$

  for all polynomials $p$ of degree $n-1$ or less. Assuming any results about Legendre polynomials that you need, prove the following results:

  (i) $\int_{-1}^{1} p(t) d t=\sum_{j=1}^{n} A_{j} p\left(x_{j}\right)$ for all polynomials $p$ of degree $2 n-1$ or less;

  (ii) $A_{j} \geqslant 0$ for all $1 \leqslant j \leqslant n$;

  (iii) $\sum_{j=1}^{n} A_{j}=2$.

  Now consider $Q_{n}(f)=\sum_{j=1}^{n} A_{j} f\left(x_{j}\right)$. Show that

  $$Q_{n}(f) \rightarrow \int_{-1}^{1} f(t) d t$$

  as $n \rightarrow \infty$ for all continuous functions $f$.

  comment

• Paper 1, Section I, I

  Topics in Analysis

  Part II, 2015

  Let $\Omega$ be a non-empty bounded open subset of $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$. Let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be continuous with $\phi$ twice differentiable on $\Omega$.

  (i) Why does $\phi$ have a maximum on $\bar{\Omega}$ ?

  (ii) If $\epsilon>0$ and $\nabla^{2} \phi \geqslant \epsilon$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

  (iii) If $\nabla^{2} \phi \geqslant 0$ on $\Omega$, show that $\phi$ has a maximum on $\partial \Omega$.

  (iv) If $\nabla^{2} \phi=0$ on $\Omega$ and $\phi=0$ on $\partial \Omega$, show that $\phi=0$ on $\bar{\Omega}$.

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• Paper 2, Section II,

  Topics in Analysis

  Part II, 2015

  State and prove Sperner's lemma concerning the colouring of triangles.

  Deduce a theorem, to be stated clearly, on retractions to the boundary of a disc.

  State Brouwer's fixed point theorem for a disc and sketch a proof of it.

  Let $g: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$ be a continuous function such that for some $K>0$ we have $\|g(x)-x\| \leqslant K$ for all $x \in \mathbb{R}^{2}$. Show that $g$ is surjective.

  comment

• Paper 3, Section II, I

  Topics in Analysis

  Part II, 2015

  Let $\alpha>0$. By considering the set $E_{m}$ consisting of those $f \in C([0,1])$ for which there exists an $x \in[0,1]$ with $|f(x+h)-f(x)| \leqslant m|h|^{\alpha}$ for all $x+h \in[0,1]$, or otherwise, give a Baire category proof of the existence of continuous functions $f$ on $[0,1]$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  at each $x \in[0,1]$.

  Are the following statements true? Give reasons.

  (i) There exists an $f \in C([0,1])$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  for each $x \in[0,1]$ and each $\alpha>0$.

  (ii) There exists an $f \in C([0,1])$ such that

  $$\limsup _{h \rightarrow 0}|h|^{-\alpha}|f(x+h)-f(x)|=\infty$$

  for each $x \in[0,1]$ and each $\alpha \geqslant 0$.

  comment

• Paper 1, Section I, H

  Topics in Analysis

  Part II, 2016

  By considering the function $\mathbb{R}^{n+1} \rightarrow \mathbb{R}$ defined by

  $$R\left(a_{0}, \ldots, a_{n}\right)=\sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right|$$

  or otherwise, show that there exist $K_{n}>0$ and $\delta_{n}>0$ such that

  $$K_{n} \sum_{j=0}^{n}\left|a_{j}\right| \geqslant \sup _{t \in[-1,1]}\left|\sum_{j=0}^{n} a_{j} t^{j}\right| \geqslant \delta_{n} \sum_{j=0}^{n}\left|a_{j}\right|$$

  for all $a_{j} \in \mathbb{R}, 0 \leqslant j \leqslant n$.

  Show, quoting carefully any theorems you use, that we must have $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$.

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• Paper 2, Section I, H

  Topics in Analysis

  Part II, 2016

  Define what it means for a subset $E$ of $\mathbb{R}^{n}$ to be convex. Which of the following statements about a convex set $E$ in $\mathbb{R}^{n}$ (with the usual norm) are always true, and which are sometimes false? Give proofs or counterexamples as appropriate.

  (i) The closure of $E$ is convex.

  (ii) The interior of $E$ is convex.

  (iii) If $\alpha: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is linear, then $\alpha(E)$ is convex.

  (iv) If $f: \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is continuous, then $f(E)$ is convex.

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• Paper 3, Section I, $2 \mathrm{H}$

  Topics in Analysis

  Part II, 2016

  In the game of 'Chicken', $A$ and $B$ drive fast cars directly at each other. If they both swerve, they both lose 10 status points; if neither swerves, they both lose 100 status points. If one swerves and the other does not, the swerver loses 20 status points and the non-swerver gains 40 status points. Find all the pairs of probabilistic strategies such that, if one driver maintains their strategy, it is not in the interest of the other to change theirs.

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• Paper 4, Section I, H

  Topics in Analysis

  Part II, 2016

  Let $a_{0}, a_{1}, a_{2}, \ldots$ be integers such that there exists an $M$ with $M \geqslant\left|a_{n}\right|$ for all $n$. Show that, if infinitely many of the $a_{n}$ are non-zero, then $\sum_{n=0}^{\infty} \frac{a_{n}}{n !}$ is an irrational number.

  comment

• Paper 2, Section II, H

  Topics in Analysis

  Part II, 2016

  Prove Bernstein's theorem, which states that if $f:[0,1] \rightarrow \mathbb{R}$ is continuous and

  $$f_{m}(t)=\sum_{r=0}^{m}\left(\begin{array}{c} m \\ r \end{array}\right) f(r / m) t^{r}(1-t)^{m-r}$$

  then $f_{m}(t) \rightarrow f(t)$ uniformly on $[0,1]$. [Theorems from probability theory may be used without proof provided they are clearly stated.]

  Deduce Weierstrass's theorem on polynomial approximation for any closed interval.

  Proving any results on Chebyshev polynomials that you need, show that, if $g:[0, \pi] \rightarrow \mathbb{R}$ is continuous and $\epsilon>0$, then we can find an $N \geqslant 0$ and $a_{j} \in \mathbb{R}$, for $0 \leqslant j \leqslant N$, such that

  $$\left|g(t)-\sum_{j=0}^{N} a_{j} \cos j t\right| \leqslant \epsilon$$

  for all $t \in[0, \pi]$. Deduce that $\int_{0}^{\pi} g(t) \cos n t d t \rightarrow 0$ as $n \rightarrow \infty$.

  comment

• Paper 4, Section II, H

  Topics in Analysis

  Part II, 2016

  Explain briefly how a positive irrational number $x$ gives rise to a continued fraction

  $$a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

  with the $a_{j}$ non-negative integers and $a_{j} \geqslant 1$ for $j \geqslant 1$.

  Show that, if we write

  $$\left(\begin{array}{ll} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$$

  then

  $$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}$$

  for $n \geqslant 0$.

  Use the observation [which need not be proved] that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1}$ to show that

  $$\left|p_{n} / q_{n}-x\right| \leqslant 1 / q_{n} q_{n+1}$$

  Show that $q_{n} \geqslant F_{n}$ where $F_{n}$ is the $n$th Fibonacci number (thus $F_{0}=F_{1}=1$, $\left.F_{n+2}=F_{n+1}+F_{n}\right)$, and conclude that

  $$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{F_{n} F_{n+1}}$$

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• Paper 1, Section I, $2 F$

  Topics in Analysis

  Part II, 2018

  State and prove Sperner's lemma concerning colourings of points in a triangular grid.

  Suppose that $\triangle$ is a non-degenerate closed triangle with closed edges $\alpha_{1}, \alpha_{2}$ and $\alpha_{3}$. Show that we cannot find closed sets $A_{j}$ with $A_{j} \supseteq \alpha_{j}$, for $j=1,2,3$, such that

  $$\bigcup_{j=1}^{3} A_{j}=\Delta, \text { but } \bigcap_{j=1}^{3} A_{j}=\emptyset$$

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• Paper 2, Section I, $2 F$

  Topics in Analysis

  Part II, 2018

  For $\mathbf{x} \in \mathbb{R}^{n}$ we write $\mathbf{x}=\left(x_{1}, x_{2}, \ldots, x_{n}\right)$. Define

  $$P:=\left\{\mathbf{x} \in \mathbb{R}^{n}: x_{j} \geqslant 0 \text { for } 1 \leqslant j \leqslant n\right\} .$$

  (a) Suppose that $L$ is a convex subset of $P$, that $(1,1, \ldots, 1) \in L$ and that $\prod_{j=1}^{n} x_{j} \leqslant 1$ for all $\mathbf{x} \in L$. Show that $\sum_{j=1}^{n} x_{j} \leqslant n$ for all $\mathbf{x} \in L$.

  (b) Suppose that $K$ is a non-empty closed bounded convex subset of $P$. Show that there is a $\mathbf{u} \in K$ such that $\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} u_{j}$ for all $\mathbf{x} \in K$. If $u_{j} \neq 0$ for each $j$ with $1 \leqslant j \leqslant n$, show that

  $$\sum_{j=1}^{n} \frac{x_{j}}{u_{j}} \leqslant n$$

  for all $\mathbf{x} \in K$, and that $\mathbf{u}$ is unique.

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• Paper 3, Section I, $2 F$

  Topics in Analysis

  Part II, 2018

  State a version of the Baire category theorem and use it to prove the following result:

  If $f: \mathbb{C} \rightarrow \mathbb{C}$ is analytic, but not a polynomial, then there exists a point $z_{0} \in \mathbb{C}$ such that each coefficient of the Taylor series of $f$ at $z_{0}$ is non-zero.

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• Paper 4, Section I, $2 F$

  Topics in Analysis

  Part II, 2018

  Let $0 \leqslant \alpha<1$ and $A>0$. If we have an infinite sequence of integers $m_{n}$ with $1 \leqslant m_{n} \leqslant A n^{\alpha}$, show that

  $$\sum_{n=1}^{\infty} \frac{m_{n}}{n !}$$

  is irrational.

  Does the result remain true if the $m_{n}$ are not restricted to integer values? Justify your answer.

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• Paper 2, Section II, F

  Topics in Analysis

  Part II, 2018

  (a) Give Bernstein's probabilistic proof of Weierstrass's theorem.

  (b) Are the following statements true or false? Justify your answer in each case.

  (i) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging pointwise to $f$ on $\mathbb{R}$.

  (ii) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $\mathbb{R}$.

  (iii) If $f:(0,1] \rightarrow \mathbb{R}$ is continuous and bounded, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $(0,1]$.

  (iv) If $f:[0,1] \rightarrow \mathbb{R}$ is continuous and $x_{1}, x_{2}, \ldots, x_{m}$ are distinct points in $[0,1]$, then there exists a sequence of polynomials $P_{n}$ with $P_{n}\left(x_{j}\right)=f\left(x_{j}\right)$, for $j=1, \ldots, m$, converging uniformly to $f$ on $[0,1]$.

  (v) If $f:[0,1] \rightarrow \mathbb{R}$ is $m$ times continuously differentiable, then there exists a sequence of polynomials $P_{n}$ such that $P_{n}^{(r)} \rightarrow f^{(r)}$ uniformly on $[0,1]$ for each $r=0, \ldots, m$.

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• Paper 4, Section II, F

  Topics in Analysis

  Part II, 2018

  We work in $\mathbb{C}$. Consider

  $$K=\{z:|z-2| \leqslant 1\} \cup\{z:|z+2| \leqslant 1\}$$

  and

  $$\Omega=\{z:|z-2|<3 / 2\} \cup\{z:|z+2|<3 / 2\}$$

  Show that if $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there is a sequence of polynomials $p_{n}$ such that $p_{n}(z) \rightarrow f(z)$ uniformly on $K$.

  Show that there is a sequence of polynomials $P_{n}$ such that $P_{n}(z) \rightarrow 0$ uniformly for $|z-2| \leqslant 1$ and $P_{n}(z) \rightarrow 1$ uniformly for $|z+2| \leqslant 1$.

  Give two disjoint non-empty bounded closed sets $K_{1}$ and $K_{2}$ such that there does not exist a sequence of polynomials $Q_{n}$ with $Q_{n}(z) \rightarrow 0$ uniformly on $K_{1}$ and $Q_{n}(z) \rightarrow 1$ uniformly on $K_{2}$. Justify your answer.

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• Paper 4, Section I, H

  Topics in Analysis

  Part II, 2019

  Show that $\pi$ is irrational. [Hint: consider the functions $f_{n}:[0, \pi] \rightarrow \mathbb{R}$ given by $\left.f_{n}(x)=x^{n}(\pi-x)^{n} \sin x .\right]$

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• Paper 3, Section I, H

  Topics in Analysis

  Part II, 2019

  State Nash's theorem for a non zero-sum game in the case of two players with two choices.

  The role playing game Tixerb involves two players. Before the game begins, each player $i$ chooses a $p_{i}$ with $0 \leqslant p_{i} \leqslant 1$ which they announce. They may change their choice as many times as they wish, but, once the game begins, no further changes are allowed. When the game starts, player $i$ becomes a Dark Lord with probability $p_{i}$ and a harmless peasant with probability $1-p_{i}$. If one player is a Dark Lord and the other a peasant the Lord gets 2 points and the peasant $-2$. If both are peasants they get 1 point each, if both Lords they get $-U$ each. Show that there exists a $U_{0}$, to be found, such that, if $U>U_{0}$ there will be three choices of $\left(p_{1}, p_{2}\right)$ for which neither player can increase the expected value of their outcome by changing their choice unilaterally, but, if $U_{0}>U$, there will only be one. Find the appropriate $\left(p_{1}, p_{2}\right)$ in each case.

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• Paper 2, Section I, H

  Topics in Analysis

  Part II, 2019

  Let $\mathcal{K}$ be the collection of non-empty closed bounded subsets of $\mathbb{R}^{n}$.

  (a) Show that, if $A, B \in \mathcal{K}$ and we write

  $$A+B=\{a+b: a \in A, b \in B\}$$

  then $A+B \in \mathcal{K}$.

  (b) Show that, if $K_{n} \in \mathcal{K}$, and

  $$K_{1} \supseteq K_{2} \supseteq K_{3} \supseteq \cdots$$

  then $K:=\bigcap_{n=1}^{\infty} K_{n} \in \mathcal{K}$.

  (c) Assuming the result that

  $$\rho(A, B)=\sup _{a \in A} \inf _{b \in B}|a-b|+\sup _{b \in B} \inf _{a \in A}|a-b|$$

  defines a metric on $\mathcal{K}$ (the Hausdorff metric), show that if $K_{n}$ and $K$ are as in part (b), then $\rho\left(K_{n}, K\right) \rightarrow 0$ as $n \rightarrow \infty$.

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• Paper 1, Section I, H

  Topics in Analysis

  Part II, 2019

  Let $T_{n}$ be the $n$th Chebychev polynomial. Suppose that $\gamma_{i}>0$ for all $i$ and that $\sum_{i=1}^{\infty} \gamma_{i}$ converges. Explain why $f=\sum_{i=1}^{\infty} \gamma_{i} T_{3^{i}}$ is a well defined continuous function on $[-1,1]$.

  Show that, if we take $P_{n}=\sum_{i=1}^{n} \gamma_{i} T_{3^{i}}$, we can find points $x_{k}$ with

  $$-1 \leqslant x_{0}<x_{1}<\ldots<x_{3^{n+1}} \leqslant 1$$

  such that $f\left(x_{k}\right)-P_{n}\left(x_{k}\right)=(-1)^{k+1} \sum_{i=n+1}^{\infty} \gamma_{i}$ for each $k=0,1, \ldots, 3^{n+1}$.

  Suppose that $\delta_{n}$ is a decreasing sequence of positive numbers and that $\delta_{n} \rightarrow 0$ as $n \rightarrow \infty$. Stating clearly any theorem that you use, show that there exists a continuous function $f$ with

  $$\sup _{t \in[-1,1]}|f(t)-P(t)| \geqslant \delta_{n}$$

  for all polynomials $P$ of degree at most $n$ and all $n \geqslant 1$.

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• Paper 2, Section II, H

  Topics in Analysis

  Part II, 2019

  Throughout this question $I$ denotes the closed interval $[-1,1]$.

  (a) For $n \in \mathbb{N}$, consider the $2 n+1$ points $r / n \in I$ with $r \in \mathbb{Z}$ and $-n \leqslant r \leqslant n$. Show that, if we colour them red or green in such a way that $-1$ and 1 are coloured differently, there must be two neighbouring points of different colours.

  (b) Deduce from part (a) that, if $I=A \cup B$ with $A$ and $B$ closed, $-1 \in A$ and $1 \in B$, then $A \cap B \neq \emptyset$.

  (c) Deduce from part (b) that there does not exist a continuous function $f: I \rightarrow \mathbb{R}$ with $f(t) \in\{-1,1\}$ for all $t \in I$ and $f(-1)=-1, f(1)=1$.

  (d) Deduce from part (c) that if $f: I \rightarrow I$ is continuous then there exists an $x \in I$ with $f(x)=x$.

  (e) Deduce the conclusion of part (c) from the conclusion of part (d).

  (f) Deduce the conclusion of part (b) from the conclusion of part (c).

  (g) Suppose that we replace $I$ wherever it occurs by the unit circle

  $$C=\{z \in \mathbb{C}|| z \mid=1\}$$

  Which of the conclusions of parts (b), (c) and (d) remain true? Give reasons.

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• Paper 4, Section II, H

  Topics in Analysis

  Part II, 2019

  (a) Suppose that $K \subset \mathbb{C}$ is a non-empty subset of the square $\{x+i y: x, y \in(-1,1)\}$ and $f$ is analytic in the larger square $\{x+i y: x, y \in(-1-\delta, 1+\delta)\}$ for some $\delta>0$. Show that $f$ can be uniformly approximated on $K$ by polynomials.

  (b) Let $K$ be a closed non-empty proper subset of $\mathbb{C}$. Let $\Lambda$ be the set of $\lambda \in \mathbb{C} \backslash K$ such that $g_{\lambda}(z)=(z-\lambda)^{-1}$ can be approximated uniformly on $K$ by polynomials and let $\Gamma=\mathbb{C} \backslash(K \cup \Lambda)$. Show that $\Lambda$ and $\Gamma$ are open. Is it always true that $\Lambda$ is non-empty? Is it always true that, if $K$ is bounded, then $\Gamma$ is empty? Give reasons.

  [No form of Runge's theorem may be used without proof.]

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• Paper 1, Section I, $\mathbf{2 H}$

  Topics in Analysis

  Part II, 2020

  Let $\gamma:[0,1] \rightarrow \mathbb{C}$ be a continuous map never taking the value 0 and satisfying $\gamma(0)=\gamma(1)$. Define the degree (or winding number) $w(\gamma ; 0)$ of $\gamma$ about 0 . Prove the following.

  (i) If $\delta:[0,1] \rightarrow \mathbb{C} \backslash\{0\}$ is a continuous map satisfying $\delta(0)=\delta(1)$, then the winding number of the product $\gamma \delta$ is given by $w(\gamma \delta ; 0)=w(\gamma ; 0)+w(\delta ; 0)$.

  (ii) If $\sigma:[0,1] \rightarrow \mathbb{C}$ is continuous, $\sigma(0)=\sigma(1)$ and $|\sigma(t)|<|\gamma(t)|$ for each $0 \leqslant t \leqslant 1$, then $w(\gamma+\sigma ; 0)=w(\gamma ; 0)$.

  (iii) Let $D=\{z \in \mathbb{C}:|z| \leqslant 1\}$ and let $f: D \rightarrow \mathbb{C}$ be a continuous function with $f(z) \neq 0$ whenever $|z|=1$. Define $\alpha:[0,1] \rightarrow \mathbb{C}$ by $\alpha(t)=f\left(e^{2 \pi i t}\right)$. Then if $w(\alpha ; 0) \neq 0$, there must exist some $z \in D$, such that $f(z)=0$. [It may help to define $F(s, t):=f\left(s e^{2 \pi i t}\right)$. Homotopy invariance of the winding number may be assumed.]

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• Paper 2, Section I, $2 \mathrm{H}$

  Topics in Analysis

  Part II, 2020

  Show that every Legendre polynomial $p_{n}$ has $n$ distinct roots in $[-1,1]$, where $n$ is the degree of $p_{n}$.

  Let $x_{1}, \ldots, x_{n}$ be distinct numbers in $[-1,1]$. Show that there are unique real numbers $A_{1}, \ldots, A_{n}$ such that the formula

  $$\int_{-1}^{1} P(t) d t=\sum_{i=1}^{n} A_{i} P\left(x_{i}\right)$$

  holds for every polynomial $P$ of degree less than $n$.

  Now suppose that the above formula in fact holds for every polynomial $P$ of degree less than $2 n$. Show that then $x_{1}, \ldots, x_{n}$ are the roots of $p_{n}$. Show also that $\sum_{i=1}^{n} A_{i}=2$ and that all $A_{i}$ are positive.

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• Paper 3, Section I, H

  Topics in Analysis

  Part II, 2020

  State Runge's theorem about the uniform approximation of holomorphic functions by polynomials.

  Explicitly construct, with a brief justification, a sequence of polynomials which converges uniformly to $1 / z$ on the semicircle $\{z:|z|=1, \operatorname{Re}(z) \leqslant 0\}$.

  Does there exist a sequence of polynomials converging uniformly to $1 / z$ on $\{z:|z|=1, z \neq 1\}$ ? Give a justification.

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• Paper 4, Section I, $2 \mathrm{H}$

  Topics in Analysis

  Part II, 2020

  Define what is meant by a nowhere dense set in a metric space. State a version of the Baire Category theorem.

  Let $f:[1, \infty) \rightarrow \mathbb{R}$ be a continuous function such that $f(n x) \rightarrow 0$ as $n \rightarrow \infty$ for every fixed $x \geqslant 1$. Show that $f(t) \rightarrow 0$ as $t \rightarrow \infty$.

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• Paper 2, Section II, H

  Topics in Analysis

  Part II, 2020

  Let $T$ be a (closed) triangle in $\mathbb{R}^{2}$ with edges $I, J, K$. Let $A, B, C$, be closed subsets of $T$, such that $I \subset A, J \subset B, K \subset C$ and $T=A \cup B \cup C$. Prove that $A \cap B \cap C$ is non-empty.

  Deduce that there is no continuous map $f: D \rightarrow \partial D$ such that $f(p)=p$ for all $p \in \partial D$, where $D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2} \leqslant 1\right\}$ is the closed unit disc and $\partial D=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}=1\right\}$ is its boundary.

  Let now $\alpha, \beta, \gamma \subset \partial D$ be three closed arcs, each arc making an angle of $2 \pi / 3$ (in radians) in $\partial D$ and $\alpha \cup \beta \cup \gamma=\partial D$. Let $P, Q$ and $R$ be open subsets of $D$, such that $\alpha \subset P$, $\beta \subset Q$ and $\gamma \subset R$. Suppose that $P \cup Q \cup R=D$. Show that $P \cap Q \cap R$ is non-empty. [You may assume that for each closed bounded subset $K \subset \mathbb{R}^{2}, d(x, K)=\min \{\|x-y\|: y \in K\}$ defines a continuous function on $\mathbb{R}^{2}$.]

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• Paper 4, Section II, H

  Topics in Analysis

  Part II, 2020

  (a) State Liouville's theorem on the approximation of algebraic numbers by rationals.

  (b) Let $\left(a_{n}\right)_{n=0}^{\infty}$ be a sequence of positive integers and let

  $$\alpha=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\ldots}}}$$

  be the value of the associated continued fraction.

  (i) Prove that the $n$th convergent $p_{n} / q_{n}$ satisfies

  $$\left|\alpha-\frac{p_{n}}{q_{n}}\right| \leqslant\left|\alpha-\frac{p}{q}\right|$$

  for all the rational numbers $\frac{p}{q}$ such that $0<q \leqslant q_{n}$.

  (ii) Show that if the sequence $\left(a_{n}\right)$ is bounded, then one can choose $c>0$ (depending only on $\alpha$ ), so that for every rational number $\frac{a}{b}$,

  $$\left|\alpha-\frac{a}{b}\right|>\frac{c}{b^{2}}$$

  (iii) Show that if the sequence $\left(a_{n}\right)$ is unbounded, then for each $c>0$ there exist infinitely many rational numbers $\frac{a}{b}$ such that

  $$\left|\alpha-\frac{a}{b}\right|<\frac{c}{b^{2}}$$

  [You may assume without proof the relation

  $$\left.\left(\begin{array}{ll} p_{n+1} & p_{n} \\ q_{n+1} & q_{n} \end{array}\right)=\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n+1} \end{array}\right)\left(\begin{array}{cc} a_{n+1} & 1 \\ 1 & 0 \end{array}\right), \quad n=1,2, \ldots .\right]$$

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• Paper 1, Section I, 2H

  Topics in Analysis

  Part II, 2021

  Write

  $$P=\left\{\mathbf{x} \in \mathbb{R}^{n}: x_{j} \geqslant 0 \text { for all } 1 \leqslant j \leqslant n\right\}$$

  and suppose that $K$ is a non-empty, closed, convex and bounded subset of $\mathbb{R}^{n}$ with $K \cap \operatorname{Int} P \neq \emptyset$. By taking logarithms, or otherwise, show that there is a unique $\mathbf{x}^{*} \in K \cap P$ such that

  $$\prod_{j=1}^{n} x_{j} \leqslant \prod_{j=1}^{n} x_{j}^{*}$$

  for all $\mathbf{x} \in K \cap P$.

  Show that $\sum_{j=1}^{n} \frac{x_{j}}{x_{j}^{*}} \leqslant n$ for all $\mathbf{x} \in K \cap P$.

  Identify the point $\mathbf{x}^{*}$ in the case that $K$ has the property

  $$\left(x_{1}, x_{2}, \ldots, x_{n-1}, x_{n}\right) \in K \Rightarrow\left(x_{2}, x_{3}, \ldots, x_{n}, x_{1}\right) \in K$$

  and justify your answer.

  Show that, given any $\mathbf{a} \in \operatorname{Int} P$, we can find a set $K$, as above, with $\mathbf{x}^{*}=\mathbf{a}$.

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• Paper 2, Section I, $2 H$

  Topics in Analysis

  Part II, 2021

  Let $\Omega$ be a non-empty bounded open set in $\mathbb{R}^{2}$ with closure $\bar{\Omega}$ and boundary $\partial \Omega$ and let $\phi: \bar{\Omega} \rightarrow \mathbb{R}$ be a continuous function. Give a proof or a counterexample for each of the following assertions.

  (i) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})>0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

  (ii) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})<0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

  (iii) If $\phi$ is four times differentiable on $\Omega$ with

  $$\frac{\partial^{4} \phi}{\partial x^{4}}(\mathbf{x})+\frac{\partial^{4} \phi}{\partial y^{4}}(\mathbf{x})>0$$

  for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

  (iv) If $\phi$ is twice differentiable on $\Omega$ with $\nabla^{2} \phi(\mathbf{x})=0$ for all $\mathbf{x} \in \Omega$, then there exists an $\mathbf{x}_{0} \in \partial \Omega$ with $\phi\left(\mathbf{x}_{0}\right) \geqslant \phi(\mathbf{x})$ for all $\mathbf{x} \in \bar{\Omega}$.

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• Paper 3 , Section I, $2 H$

  Topics in Analysis

  Part II, 2021

  State Runge's theorem on the approximation of analytic functions by polynomials.

  Let $\Omega=\{z \in \mathbb{C}, \operatorname{Re} z>0, \operatorname{Im} z>0\}$. Establish whether the following statements are true or false by giving a proof or a counterexample in each case.

  (i) If $f: \Omega \rightarrow \mathbb{C}$ is the uniform limit of a sequence of polynomials $P_{n}$, then $f$ is a polynomial.

  (ii) If $f: \Omega \rightarrow \mathbb{C}$ is analytic, then there exists a sequence of polynomials $P_{n}$ such that for each integer $r \geqslant 0$ and each $z \in \Omega$ we have $P_{n}^{(r)}(z) \rightarrow f^{(r)}(z)$.

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• Paper 4, Section I, 2H

  Topics in Analysis

  Part II, 2021

  (a) State Brouwer's fixed-point theorem in 2 dimensions.

  (b) State an equivalent theorem on retraction and explain (without detailed calculations) why it is equivalent.

  (c) Suppose that $A$ is a $3 \times 3$ real matrix with strictly positive entries. By defining an appropriate function $f: \triangle \rightarrow \triangle$, where

  $$\triangle=\left\{\mathbf{x} \in \mathbb{R}^{3}: x_{1}+x_{2}+x_{3}=1, x_{1}, x_{2}, x_{3} \geqslant 0\right\}$$

  show that $A$ has a strictly positive eigenvalue.

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• Paper 2, Section II, H

  Topics in Analysis

  Part II, 2021

  Let $r:[-1,1] \rightarrow \mathbb{R}$ be a continuous function with $r(x)>0$ for all but finitely many values of $x$.

  (a) Show that

  $$\langle u, v\rangle=\int_{-1}^{1} u(x) v(x) r(x) d x$$

  defines an inner product on $C([-1,1])$.

  (b) Show that for each $n$ there exists a polynomial $P_{n}$ of degree exactly $n$ which is orthogonal, with respect to the inner product $(*)$, to all polynomials of lower degree.

  (c) Show that $P_{n}$ has $n$ simple zeros $\omega_{1}(n), \omega_{2}(n), \ldots, \omega_{n}(n)$ on $[-1,1]$.

  (d) Show that for each $n$ there exist unique real numbers $A_{j}(n), 1 \leqslant j \leqslant n$, such that whenever $Q$ is a polynomial of degree at most $2 n-1$,

  $$\int_{-1}^{1} Q(x) r(x) d x=\sum_{j=1}^{n} A_{j}(n) Q\left(\omega_{j}(n)\right)$$

  (e) Show that

  $$\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right) \rightarrow \int_{-1}^{1} f(x) r(x) d x$$

  as $n \rightarrow \infty$ for all $f \in C([-1,1])$.

  (f) If $R>1, K>0, a_{m}$ is real with $\left|a_{m}\right| \leqslant K R^{-m}$ and $f(x)=\sum_{m=1}^{\infty} a_{m} x^{m}$, show that

  $$\left|\int_{-1}^{1} f(x) r(x) d x-\sum_{j=1}^{n} A_{j}(n) f\left(\omega_{j}(n)\right)\right| \leqslant \frac{2 K R^{-2 n+1}}{R-1} \int_{-1}^{1} r(x) d x$$

  (g) If $r(x)=\left(1-x^{2}\right)^{1 / 2}$ and $P_{n}(0)=1$, identify $P_{n}$ (giving brief reasons) and the $\omega_{j}(n)$. [Hint: A change of variable may be useful.]

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• Paper 4, Section II, H

  Topics in Analysis

  Part II, 2021

  Let $x$ be irrational with $n$th continued fraction convergent

  $$\frac{p_{n}}{q_{n}}=a_{0}+\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}}$$

  Show that

  $$\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} a_{0} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{1} & 1 \\ 1 & 0 \end{array}\right) \cdots\left(\begin{array}{cc} a_{n-1} & 1 \\ 1 & 0 \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right)$$

  and deduce that

  $$\left|\frac{p_{n}}{q_{n}}-x\right| \leqslant \frac{1}{q_{n} q_{n+1}} .$$

  [You may quote the result that $x$ lies between $p_{n} / q_{n}$ and $p_{n+1} / q_{n+1} \cdot$ ]

  We say that $y$ is a quadratic irrational if it is an irrational root of a quadratic equation with integer coefficients. Show that if $y$ is a quadratic irrational, we can find an $M>0$ such that

  $$\left|\frac{p}{q}-y\right| \geqslant \frac{M}{q^{2}}$$

  for all integers $p$ and $q$ with $q>0$.

  Using the hypotheses and notation of the first paragraph, show that if the sequence $\left(a_{n}\right)$ is unbounded, $x$ cannot be a quadratic irrational.

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