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Let $f$ be a continuous function on $\mathbb{R}$ with compact support and unique maximum. Form the functions $$ F_{n,k}(x)=f^n\left(x-\frac{k}{2^n}\right), k \in Z, n>0 $$

I am wondering if one can expand a Hermite function in terms of $F_{n,k}$, i.e. something like $\sum_{k,n}c_kF_{n,k} $

Note: $ h_n(x)=(-1)^n\gamma_ne^{x^2/2} \frac{d^n}{dx^n}e^{-x^2}, $ where $\gamma_n=\pi^{-1/4}2^{-n/2}(n!)^{-1/2}$, is a Hermite function.

Thank you.

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  • $\begingroup$ Is this homework, as some of your previous questions on MSE have been? If not, why are you looking to prove this particular result? $\endgroup$
    – Yemon Choi
    Commented Jun 17, 2012 at 11:37
  • $\begingroup$ No, it is not a homework. In fact, my previous questions wasn't either.there was just an ideas to prove in different way some homework questions plus personal interes. The reason for my current question- I would like to study the properties of the $F_{n,k}(x)$ (if $F_{n,k}$ can form Riesz basis, Bessel system, so on). $\endgroup$
    – David
    Commented Jun 17, 2012 at 12:51
  • $\begingroup$ In what sense do you want your sum to converge? $\endgroup$
    – Yemon Choi
    Commented Jun 18, 2012 at 19:29
  • $\begingroup$ I would like to get $L_p$ convergense. At least for $p=2$. $\endgroup$
    – David
    Commented Jun 19, 2012 at 18:34

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