Zernike functions, soft adaptive optics and rigged Hilbert spaces.

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1,1) x su(1,1). We show how they induce discrete as well continuous bases that coexist in the framework of rigged Hilbert spaces. We also discuss some other interesting properties of Zernike polynomials and Zernike functions. One of the interests of Zernike functions has been their applications in optics. Here, we show how operators on the spaces of Zernike functions play a role in optical image processing. We call this procedure "soft adaptive optics" to distinguish it from the standard adaptive optics as is currently used.