Approximation via gradients on the ball. The Zernike case

In this work, we deal in a d dimensional unit ball equipped with an inner product constructed by adding a mass point at zero to the classical ball inner product applied to the gradients of the functions. Apart from determining an explicit orthogonal polynomial basis, we study approximation properties of Fourier expansions in terms of this basis. In particular, we deduce relations between the partial Fourier sums in terms of the new orthogonal polynomials and the partial Fourier sums in terms of the classical ball polynomials. We also give an estimate of the approximation error by polynomials of degree at most n in the corresponding Sobolev space, proving that we can approximate a function by using its gradient. Numerical examples are given to illustrate the approximation behavior of the Sobolev basis.