Friday, 10 March 2017

analysis - How to show this series has at most three nonzero consecutive terms and at least one not null?

Let $\psi\in C^\infty_0(\mathbb R^n)$ such that $1\leq \psi\leq 0$ and satisfying $$\psi(x)=\left\{\begin{array}{lcl}1&\textrm{if}&1\leq |x|\leq 2\\0&\textrm{if}&|x|<1/2\ \textrm{or}\ |x|>4 \end{array}\right.$$ For $k=1, 2, \ldots$ define $$\psi_k(\xi)=\psi\left(\frac{\xi}{2^{k-1}}\right),$$ and set $$\Phi(\xi)=\sum_{k=0}^\infty \psi_k(\xi),$$ where $\psi_0\in C^\infty_0(\mathbb R^n)$ is such that $0\leq \psi_0(\xi)\leq 1$ and $$\psi_0(\xi)=\left\{\begin{array}{lcl}
1&\textrm{se}& |\xi|\leq 1\\ 0&\textrm{if}&|\xi|>2\end{array}\right.$$



Question: How can I show that for every $\xi\in\mathbb R^n$ fixed the series which defines $\Phi(\xi)$ has at most three nonzero consecutive terms and at least one is non zero?



Obs: Notice $$\textrm{supp}(\psi_k)\subset \{\xi\in\mathbb R^n: 2^{k-2}\leq |\xi|\leq 2^{k+1}\}.$$




Any help will be valuable. Thanks.



Notation: $C^\infty_0(\mathbb R^n)$ consists of those $C^\infty$ functions on $\mathbb R^n$ with compact support.

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