Zernike functions, rigged Hilbert spaces, and potential applications

We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1, 1) ⊕ su(1, 1). We show how they induce discrete as well as 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 areas of interest of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing.We revise the symmetries of the Zernike polynomials that determine the Lie algebra su(1, 1) ⊕ su(1, 1). We show how they induce discrete as well as 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 areas of interest of Zernike functions has been their applications in optics. Here, we suggest that operators on the spaces of Zernike functions may play a role in optical image processing.