\((0,2)\) Pál-type interpolation on a circle in the complex plane involving Möbius transforms (Q2481412)

Let \(A\) and \(B\) are polynomials of degree \(n\) and \(m\) respectively. The \((0,2)\) Pàl-type interpolation problem on \({A(z),B(z)}\) consists to determine a polynomial \(P\) of degree at most \(N=m+n-1\) satisfying the following interpolation conditions: \[ P(y_k)=0,\quad A(y_k)=0,\quad k=1,2,\ldots,n \] \[ P''(z_k)=0,\quad B(z_k)=0,\quad k=1,2,\ldots,m \] If \(P\) is unique the problem is called regular. The authors generalize the approach of nonuniformly distributed nodes on the unit circle to the case of nodes \(y_k\) and \(z_k\) obtained by the Möbius map \(T_\alpha =\rho\rho'\frac{z-\alpha}{\rho^2-\alpha z}\), \((0<\alpha<\rho)\) of zeros of \((z^{2n}-\rho^{2n})\) and present some regularity conditions relating \(\alpha\) with \(\rho\).