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We need to find a series of functions $f_m$ verifying the property $$\int_{-\infty}^{+\infty}\! x^{2n} f_m\!(x) \;{\text d}x=\delta_{nm}.$$ We already have found $$f_0(x) = \frac{e^{-\sqrt{|x|}}\sin(\sqrt{|x|})}{|x|}$$ (within a $\pi/2$ normalisation factor) using $$\int_0^{+\infty}\!t^{4n+3} e^{-(1+i)t}\;{\text d}t \in\mathbb R, \quad \forall \; n \!\in\! \mathbb N$$ (integrate for example along the border of the bottom right quarter of the complex plane...)

If anyone can help us find the next functions of the series, that'd be great!


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I suspect there is more to your problem than you have stated: You could (trivially) consider the collection of functions $\{p_k\}_{k=0}^{\infty}$ defined by $p_k(x) = x^{2k} 1_{[0,1]}(x)$ ($1_{[0,1]}$ is the indicator function of the set $[0,1]$). Define an inner product $<a,b> = \int_{-\infty}^{+\infty} a(x)b(x) dx$, and then use a Gram-Schmidt technique to orthogonalize the $p_k$. Then scale the resulting functions so that your property holds true. – copper.hat Apr 13 '12 at 5:50
Hi, Thanks for your answer. There is nothing more to the problem except that I want a "useful" expression for those functions, not a conceptual "what you would get if you did a whole lot of nasty calculations" kind of thing... Also, if I'm not wrong, orthogonalising your $p_k$ does not guarantee that your final functions will satisfy the property. To me, there is no trivial reason why the $f_m$ should be orthogonal to each other, and as the $x^{2n}$ themselves aren't orthogonal, I suspect the $f_m$ won't be either... – Julien Apr 13 '12 at 8:17
The $f_m$ will be orthogonal by construction, that's the whole point of Gram-Schmidt orthogonalization, but that's not the issue. The problem with my suggestion is that the functions $x\mapsto x^{2n}$ only satisfy the orthogonality condition for $n<m$. – copper.hat Apr 13 '12 at 8:34

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