Problem set 15, Harmonic Analysis

Due May 31st

Problem 1

Let $latex u(x) = e^{\omega x}$ on $latex I=[0,1]$. Then, for each $latex m\ge 1$, $latex u|_{\mathcal P_m}$ is a discrete eigenfunction of $latex \Delta_m$ with eigenvalue

$latex \lambda_m = \dfrac{\omega^2}{4^m} + O(2^{-3m})$.

$latex \mathcal P_m$ is the dyadic partition $latex \{0, 1/2^m, \ldots, 1\}$ of $latex [0,1]$.

Problem 2

Let $latex \phi(x) = 2 - \sqrt{4-x}$, for $latex x\in[0,4]$.

1. $latex \phi(x) = \dfrac{1}{4}x + O(x^2)$ as $latex x\to0$.
2. The sequence defined by, for given $latex \lambda_0\in[0,2]$, $latex \lambda_m = \phi(\lambda_{m-1})$ and $latex x_m = 4^m\lambda_m$ for $latex m\ge1$ satisfies
   
   $latex x_m - x_{m-1} = O(2^{-m})$.
3. $latex x_m$ is Cauchy and hence converges.

Problem 3

Let $latex \psi(x) = \dfrac{5 - \sqrt{25- 4x}}{2}$, for $latex x\in[0,4]$.

1. $latex \psi(x) = \dfrac{1}{5}x + O(x^2)$ as $latex x\to0$.
2. The sequence defined by, for given $latex \lambda_0\in[0,2]$, $latex \lambda_m = \psi(\lambda_{m-1})$ and $latex x_m = 5^m\lambda_m$ for $latex m\ge1$ satisfies
   
   $latex x_m - x_{m-1} = O(5^{-m})$.
3. $latex x_m$ is Cauchy and hence converges.

Problem 4

The sequence $latex u_m$ of functions resulting from the decimation process on the Sierpiński gasket constructs a uniformly continuous function on $latex V_*$.

Problem 5

Calculate the eigenvalues and eigenvectors of the matrix

$latex \begin{pmatrix}4 & -1 & -1\\ -1 & 4 & -1 \\ -1 & - 1 & 4 \end{pmatrix}.$