# 2D basis functions orthogonal under exponential kernel

In one dimension, the Laguerre polynomials are orthogonal under exponential weighting: $$\int_0^\infty L_n(x) L_m(x) e^{-x} \, dx = 0, n \ne m$$ Does anyone know what the corresponding basis functions would be in 2 dimensions? $$\int_{-\infty}^\infty \int_{-\infty}^\infty F_n(x,y) F_m(x,y) e^{-r} \, dx \, dy = 0, r= \sqrt{x^2+y^2}, n \ne m$$ The Zernike polynomials are orthogonal, but with uniform weight and over the unit disk.

The underlying problem is to compute an estimator for a missing pixel. A series of orthogonal functions are helpful since you can then incrementally compute a 1-st order estimator, then a second order, then a third order, and so on. The exponential arises since (in natural scenes anyway) presumably pixels further away have less influence.

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Supposing $F_n$'s are functions of $r$, your orthogonality relation is $$\int_0^\infty F_m F_n\ r e^{-r} dr = 0$$ so $F_m$ is the generalized Laguerre polynomial $L^{(1)}_n$, see here and here. To get an orthogonal basis of $L^2(\mathbb{R}^2,e^{-r}dx\,dy)$, you need to use all the functions $L^{(1)}_n(r)\, e^{ik\phi}$.
Thanks, that makes sense. Since I have real valued data, and want real coefficients (application is biology), the functions become $L(r)cos(k\phi)$ and $L(r)sin(k\phi)$, unless I'm missing something... –  Lou Scheffer Aug 24 '12 at 21:02