  This paper studies the effects on Zernike coefficients of aperture scaling,
translation and rotation, when a given aberrated wavefront is described on the
Zernike polynomial basis. It proposes a new analytical method for computing the
matrix that enables the building of the transformed Zernike coefficients from
the original ones. The technique is based on the properties of Zernike
polynomials Fourier Transform and, in the case of a full aperture without
central obstruction, the coefficients of the matrix are given in terms of
integrals of Bessel functions. The integral formulas are exact and do not
depend on any specific ordering of the polynomials.
