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Let $$ f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+, $$ where $\alpha > -\frac 12$(see for reference http://bigwww.epfl.ch/publications/unser9901.pdf).

I am wondering if one can get nice representation of $L_2$-norm of the function $f^{\alpha}(x)$, namely $$ \int_{-\infty}^{\infty}(f^{\alpha}_+(x))^2dx. $$

(Here $y^{\alpha}_+=\max(0, y)^{\alpha}$ is a one-sided power function).

Thank you.

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$f_+^\alpha$ doesn't belong to $L^2$ –  no identity May 19 '12 at 12:59
    
It is. Function $f^{\alpha}$ is a fractional spline. It was proved by M. Uhser bigwww.epfl.ch/publications/unser9901.pdf , that it belongs to $L_2$. –  David May 19 '12 at 15:52
    
When, I tried to plot this function it turns out to tend to minus infinity. And if your question raised from this paper, why don't you put this reference in your question? –  no identity May 19 '12 at 18:40
    
What is the meaning of $\binom{\alpha+1}{k}$ and $(x-k)^{\alpha}$? –  users31526 May 19 '12 at 21:09
    
What is the meaning of "representation of $L_{2}-$norm"? –  users31526 May 19 '12 at 21:18

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