Problem set 16, Harmonic Analysis

Due June 7

Problem 1

For $latex m\ge 2$, there is no Dirichlet eigenfunction on $latex V_m$ with respect to $latex \lambda_m=2$.

Problem 2

1. The Dirichlet eigenfunctions on $latex V_m$ constructed in class, with respect to $latex \lambda_m=5$, are linearly independent.
2. If we add the three eigenfunctions chained from $latex p_1$ to $latex p_2$, from $latex p_2$ to $latex p_3$, and from $latex p_3$ to $latex p_1$, they are linearly dependent.

Problem 3

Let $latex \mathcal E_m$ be a sequence of Dirichlet forms such that 

$latex \displaystyle \mathcal E_m(u,v) = \sum_{j=1}^N \frac{1}{r_j}\mathcal E_{m-1}(u\circ f_j,v\circ f_j)$

and, given a function $latex u$ on $latex V_0$,

$latex \min\{\mathcal E_1(v,v): v|_{V_0} = u\} = \mathcal E_0(u,u)$,

then $latex \min\{\mathcal E_m(v,v): v|_{V_{m-1}} = u\} = \mathcal E_{m-1}(u,u)$ for every $latex m\ge1$ and any given function $latex u$ on $latex V_{m-1}$.

Problem 4

Calculate all harmonic structures on the interval, self-similar with contractions $latex f_1(x) = \dfrac{1}{2}x$ and $latex f_2(x) = \dfrac{1}{2}x + \frac{1}{2}$ on the real line.

Problem 5

Let $latex \mathcal E$ be a symmetric bilinear form on functions on the set $latex V$, and suppose $latex H$ is the symmetric operator that induces $latex \mathcal E$,

$latex \mathcal E(u,v) = -\langle u, Hv \rangle$.

The the following are equivalent:

1. $latex \mathcal E(u,u)\ge 0$, and $latex \mathcal E(u,u)=0$ if and only if $latex u$ is constant.
2. $latex H$ is nonpositive definite, and $latex Hu=0$ if and only if $latex u$ is constant.

(Hint: Use the spectral theorem.)