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Problem set 13, Harmonic Analysis

Due May 17

Problem 1

Let $latex s\ge 0$ and $latex \mathcal H^s$ the Hausdorff measure with exponent $latex s$ in $latex \mathbb R^d$.
  1. If $latex A\subset B$, then $latex \mathcal H^s(A) \le \mathcal H^s(B)$.
  2. If $latex A = \bigcup_j A_j$, then $latex \displaystyle \mathcal H^s(A) \le \sum_j \mathcal H^s(A_j)$.
  3. If $latex \text{dist}(A,B)>0$, then $latex \mathcal H^s(A\cup B) = \mathcal H^s(A) + \mathcal H^s(B)$.

Problem 2

If $latex A\subset\mathbb R^d$ is countable, then $latex \dim(A) = 0$.

Problem 3

If $latex 0 < p < 1$, the function $latex x\mapsto x^p$ is concave: for all $latex x,y>0$ and $latex t\in[0,1]$, 
$latex (tx + (1-t)y)^p \ge t x^p + (1-t) y^p$.

Problem 4

Let $latex f:\mathbb R^d \to \mathbb R^d$ be a similitude with coefficient $latex \alpha >0$: for every $latex x,y\in\mathbb R^d$, 
$latex |f(x) - f(y)| = \alpha |x-y|$.
Let $latex g(x) = \dfrac{1}{\alpha} (f(x) - f(0))$.
  1. For all $latex x,y\in\mathbb R^d$, $latex g(x)\cdot g(y) = x\cdot y$.
  2. If $latex e_1, e_2,\ldots,e_d$ is the standard basis for $latex \mathbb R^d$, then $latex g(e_1),g(e_2),\ldots,g(e_d)$ is an orthonormal basis for $latex \mathbb R^d$.
  3. For $latex x,y\in\mathbb R^d$ and $latex j=1,2,\ldots,d$,
    $latex g(x+y)\cdot g(e_j) = (g(x) + g(y))\cdot g(e_j)$.
  4. For $latex x\in\mathbb R^d$, $latex \lambda\in\mathbb R$ and $latex j=1,2,\ldots,d$,
    $latex g(\lambda x)\cdot g(e_j) = \lambda g(x)\cdot g(e_j)$.
  5. $latex g$ is an orthogonal linear transformation.
Conclude that every similitude in $latex \mathbb R^d$ is of the form
$latex f(x) = \alpha Mx + x_0$,
where $latex \alpha>0$, $latex M\in O(d)$ and $latex x_0\in\mathbb R^d$.

Problem 5

The polynomial $latex x^3 + 2x - 1$ has a unique real root, and the root is in $latex(0,\gamma)$, where
$latex \displaystyle \gamma = \frac{1}{\varphi} = \frac{\sqrt 5 - 1}{2}.$