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Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ One can show that $\text{span}\{f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right), k \in \mathbb{Z}, m>0\}$ is dense in $L_p(R)$.

Under which conditions does $\{f_{m,k}\}$ form a frame (or maybe a Riesz basis) for $L_p(R)$?

Thank you.

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By $f^m$ you mean the pointwise power? Are you sure about your claim for –  Dirk Jun 11 '12 at 13:45
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My objection about the lack of oscillation in $f_{m,k}$ applies to this construction as well. math.stackexchange.com/questions/155783/… –  user31373 Jun 11 '12 at 14:23
    
@Leonid Kovalev: Thank you. –  David Jun 11 '12 at 15:14
    
I just noticed tat my above comment is incomplete. I wanted to ask if you are sure about your claim $p=\infty$... –  Dirk Jun 11 '12 at 20:23
    
Yes, its for $p\ge 2$. –  David Jun 11 '12 at 21:30

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