# Finding a series of functions orthogonal to all $x^{2n}$ but one

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!

Thanks!

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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