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Let g be integrable, even function. Let $\Lambda$ be uniformly discrete sequaence (i.e. $\inf|\lambda_i-\lambda_j|>0, \lambda_i, \lambda_j \in \Lambda$).

We say that system $\Psi=\{\psi_n\}$ is $(g, \Lambda)$-localized if $|\psi_n(t)|<cg(t-\lambda_n)$, where $c>0$ some constant.

Let function $f$ be continuous on $R$ with compact support and with exactly one maximum. We form the followinf system $\{f_{m,k}\}=\left\{f^m\left(t-\frac{k}{2^m}\right)\right\}, k\in Z, m>0.$

I would like to understand if the system $\{f_{m,k}\}$ is $(g, \Lambda)$-localized?

Thank you for your help.

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Re your first sentence, there are no even decreasing functions, and the only even non-increasing functions are constant. – Amit Kumar Gupta Aug 29 '12 at 2:46
Certainly not every such system is localized for every $g$ and $\Lambda$. Just look at, say $g=0$. Do you have any more information on the form of solution you're hoping to find? – Kevin Carlson Aug 29 '12 at 3:51
Well, I am interested in the case when function $g$ is decreasing function. In particular, when it is a 'hat' function, i.e. something like Gaussian. – Michael Aug 29 '12 at 5:42

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