**5. Computation of the singular values** *λ<sup>k</sup>*

Elements of this basis are given in terms of Jacobi polynomials as

$$\phi\_k^{\nu,m}(\mathbf{z}) = \frac{(-\mathbf{1})^{\min(m,k)}}{\left(\mathbf{1} - |\mathbf{z}|^2\right)^m} |\mathbf{z}|^{|m-k|} e^{-i(m-k)\text{arg}\mathbf{z}} P\_{\min(m,k)}^{(|m-k|, \ 2(\nu-m)-1)}(\mathbf{1} - 2\mathbf{z}\overline{\mathbf{z}}).\tag{30}$$

The norm square of *ϕ<sup>ν</sup>*,*<sup>m</sup> <sup>k</sup>* in *<sup>L</sup>*2,*<sup>ν</sup>* ð Þ is given by

$$\rho\_k^{\nu, m} = \frac{\pi}{(2(\nu - m) - 1)} \frac{(m \lor k)! \Gamma(2(\nu - m) + m \land k)}{(m \land k)! \Gamma(2(\nu - m) + m \lor k)}. \tag{31}$$

Here, *m* ∧ *k* ≔ min ð Þ *m*, *k* and *m* ∨ *k* ≔ max ð Þ *m*, *k :* Let us introduce the notation. The set of functions

$$\gamma\_k^{\nu,m} := \frac{(-1)^{m \wedge k}}{\sqrt{\rho\_k^{\nu,m}}}, k = 0, 1, 2, \dots \tag{32}$$

So that we consider the elements

$$\Phi\_k^{\nu,m}(z) \coloneqq \gamma\_k^{\nu,m} \frac{1}{\left(1 - z\overline{z}\right)^m} |x|^{|m-k|} e^{-i(m-k)\text{arg}\overline{z}} P\_{\min\left(m,k\right)}^{\left(|m-k|,\ 2\left(\nu-m\right)-1\right)} \left(1 - 2z\overline{z}\right). \tag{33}$$
