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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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    Cross-posted to MO: http://mathoverflow.net/questions/97141/l-2-norm-representation2012-06-08

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