In this paper, we study and develop some properties of classical orthogonal polynomials on the unit ball, specifically those that are orthogonal with respect to the inner product: 1 \(\begin{aligned} \langle f,g\rangle _{0}=b_{0}\int _{\mathbb {B}^2}f(x)g(x) W_0(x)dx, \end{aligned}\) where the weight function \(W_{0}\) is 1, and \(b_{0}\) is a normalization constant that satisfies \(\langle 1,1 \rangle _{0}=1\) . This case is referred to as the Zernike Case. In this case, we consider various Sobolev-type inner products found in the literature and examine the special case of Zernike. Based on this, we develop the basis, succeeding in expressing them as linear combinations of the radial part of the unperturbed Zernike polynomials. We present several examples of polynomials in this context and graph the first polynomials, writing them explicitly.
