Laguerre-Gaussian functions are very common in optics and I wonder if they are Compactly Supported. These functions are essentially an associated Laguerre Polynomial modulated by a gaussian function.
Also if someone would please recommend some good bibliography on the subject because coleagues seem to misuse the term.
Laguerre-Gaussian functions, as Jyrki notes, are not compactly supported. Rather, the Laguerre polynomials form an orthogonal basis over the Hilbert space $L^2(0,\infty)$ having inner product
$$\langle f,g\rangle = \int_0^{\infty} dx \, f(x)\, g(x)\, e^{-x}$$
In optical beam profiles, you are not going to find many good representations of functions with finite support, except as an infinite sum over functions like Laguerre Gaussians that are not finitely supported.
Nonetheless, if you want a reference that treats Laguerre polynomials with all due respect, I recommend the book my professor Sam Holland wrote on Hilbert space for undergraduates: Applied Analysis by the Hilbert Space Method.