A generalization of Pal-type interpolation (Q1900724)

The paper is concerned with a generalization of the Pal-type interpolation with respect to the zeros \((\cos \theta_k )^n_{k=1}\) of the \(n\)-th Chebyshev polynomial of the first kind and the zeros \((\cos \varphi_k )^{n- 1}_{k= 1}\) of the \((n-1)\)-st Chebyshev polynomial of the second kind. In the first part of the paper it is proved that for any real numbers \((\alpha_k )^n_{k=1}\) and \((\beta_k )^{n- 1}_{k= 1}\) there exists a unique polynomial \(P_n\) of degree at most \(2n-1\) such that \[ P_n (\cos \theta_k )= \alpha_k, \quad k=1, \dots, n, \qquad {{\delta_{h/2} (P_n (\cos \varphi_k))} \over {h \sin \varphi_k}}= \beta_k, \quad k=1, \dots, n-1, \tag \(*\) \] where \(0< h< \pi/ (2n)\) and \[ \delta_{h/2} (f (\cos \theta))= f \bigl( \cos (\theta+ {\textstyle {h\over 2}}) \bigr)- f \bigl( \cos (\theta- {\textstyle {h\over 2}} )\bigr). \] The corresponding Pal-type interpolation in the limit case \(h\downarrow 0\) of this kind of interpolation. In the second part of the paper the fundamental polynomials and the rate of convergence with respect to this kind of interpolation are studied.