2
$\begingroup$

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.

  • 0
    $f_+^\alpha$ doesn't belong to $L^2$2012-05-19
  • 0
    It is. Function $f^{\alpha}$ is a fractional spline. It was proved by M. Uhser http://bigwww.epfl.ch/publications/unser9901.pdf , that it belongs to $L_2$.2012-05-19
  • 0
    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?2012-05-19
  • 0
    What is the meaning of $\binom{\alpha+1}{k}$ and $(x-k)^{\alpha}$?2012-05-19
  • 0
    What is the meaning of "representation of $L_{2}-$norm"?2012-05-19
  • 0
    For the record, I attempted to do this using the Fourier transform of $f_+^\alpha$ given in the paper (equation 3.1, page 52). Spending a couple of minutes messing around with it in Mathematica, I couldn't get a nice expression. Basically, I messed around with $$\int_{-\infty}^\infty \bigg({(1-e^{ix})(1-e^{-ix})\over x^2}\bigg)^{\alpha+1}\ dx$$ This seems like it could be tractable, but it might not be.2012-05-19
  • 0
    @Kuashik, $\big({u\atop v}\big)={\Gamma(u+1)\over\Gamma(v+1)\Gamma(u-v+1)}$ and $x^\alpha_+=\cases{x^\alpha& $x\ge 0$\cr 0& else}$ (this is defined in the OP).2012-05-19
  • 0
    Sorry B R : I know $x^n$ where $n$ is an integer, $x^{\alpha}$ for $x>0, \alpha \in \mathbb R$2012-05-19
  • 0
    @Kuashik, it's always ok to ask for definitions of things. And $x_+^\alpha$ is certainly not a standard thing, unless you've studied a bit about distributions.2012-05-19
  • 0
    Cross-posted to MO: http://mathoverflow.net/questions/97141/l-2-norm-representation2012-06-08

0 Answers 0