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Mostrando las entradas de junio, 2019

### Harmonic Analysis: Week 15 notes

Notes on the eigenfunctions of the Laplacian on the Sierpińsky gasket:  Chapter15.pdf

### Harmonic Analysis: Week 14 notes

Notes on the harmonic structure and the Laplacian on the Sierpiński gasket:  Chapter14.pdf

### Harmonic Analysis: Week 13 notes

Notes on the Hausdorff dimension and fractal sets:  Chapter13.pdf

### Problem set 16, Harmonic Analysis

Due June 7 Problem 1 For $m\ge 2$, there is no Dirichlet eigenfunction on $V_m$ with respect to $\lambda_m=2$. Problem 2 The Dirichlet eigenfunctions on $V_m$ constructed in class, with respect to $\lambda_m=5$, are linearly independent. If we add the three eigenfunctions chained from $p_1$ to $p_2$, from $p_2$ to $p_3$, and from $p_3$ to $p_1$, they are linearly dependent. Problem 3 Let $\mathcal E_m$ be a sequence of Dirichlet forms such that  $\displaystyle \mathcal E_m(u,v) = \sum_{j=1}^N \frac{1}{r_j}\mathcal E_{m-1}(u\circ f_j,v\circ f_j)$ and, given a function $u$ on $V_0$, $\min\{\mathcal E_1(v,v): v|_{V_0} = u\} = \mathcal E_0(u,u)$, then $\min\{\mathcal E_m(v,v): v|_{V_{m-1}} = u\} = \mathcal E_{m-1}(u,u)$ for every $m\ge1$ and any given function $u$ on $V_{m-1}$. Problem 4 Calculate all harmonic structures on the interval, self-