Before revision:  We prove that the convolution of two functions can be represented as a simple dot product between Zernike polynomial coefficients . We also prove that a rotation of a convolution kernel equates to a set of 2 by 2 rotation matrices applied to Zernike polynomial coefficients, which can be critical in manifold domains .
After revision:  We prove that the convolution of two functions can be represented as a simple dot product between Zernike polynomial coefficients ; and the rotation of a convolution kernel equates to a set of 2 by 2 rotation matrices applied to Zernike polynomial coefficients, which can be critical in manifold domains .