Birkhoff interpolation on unity and on Möbius transform of the roots of unity
For a nonnegative integer \(n\) and \(0<\alpha<1\) let \(z_0,z_1,\dotsc,z_{n-1}\) be the zeros of the polynomial \[ \omega_n^\alpha(z) := (z+\alpha)^n+(1+\alpha z)^n . \] Then \(z_0,z_1,\dotsc,z_{n-1}\) are the images of the zeros \(w_0,w_1,\dotsc,w_{n-1}\) of the polynomial \(z^n+1\) under the Möbius transform \[ M_\alpha(z) := \frac{z-\alpha}{1-\alpha z} , \] and they are not uniformly distributed on the unit circle. \textit{M. G.~de Bruin, A.~Sharma} and \textit{J.~Szabados} [Approximation theory. In memory of A. K. Varma (Pure Appl. Math., Marcel Dekker 212, 167-179) (1998; Zbl 0907.41001)] have shown that the Birkhoff interpolation problem \[ P^{(\nu)}(z_k) = c_{\nu,k} , \quad \nu=0,1,\dotsc,r-2,r , \quad k=0,1,\dotsc,n-1 \tag{1} \] is regular, i.e., there exists a unique polynomial \(P\) of degree \(rn-1\) satisfying (1). In the present paper, the authors prove that this remains valid, if the extra node \(z_n=1\) is added. Furthermore, the authors consider the following Pál-type interpolation problem \[ P(z_k')=c_k , \quad P'(z_k)=b_k , \quad k=0,1,\dotsc,n-1 , \quad P'(1)=d , \] where \(z_0',z_1',\dotsc,z_{n-1}'\) are the zeros of the polynomial \[ v_n^\alpha(z) := (z+\alpha)^n-(1+\alpha z)^n , \] i.e., \(z_0',z_1',\dotsc,z_{n-1}'\) are the images of the \(n\)-th roots of unity under \(M_\alpha\). It is shown that this problem is also regular.
scientific article; zbMATH DE number 1456937