Pál-type interpolation on nonuniformly distributed nodes on the unit circle
Let \(A=\{y_1, \dots, y_n\}\) and \(B=\{z_1, \dots, z_m\}\) be distinct complex numbers, and let \(r\) be a positive integer. The Pál-type \((0,r)\) interpolation problem \(\{A,B\}\) is the question whether polynomials of degree \(\leq n+m-1\) exist which are required to take arbitrary prescribed values at the points \(y_1, \dots, y_n,\) and arbitrary prescribed values of the \(r\)-th derivative at the points \(z_1,\dots,z_m\). The problem is called regular, if for every choice of values there exists a unique such polynomial. The author studies the regularity of Pál-type \((0,1)\) and \((0,2)\) interpolation problems for certain nonuniformly distributed nodes on the unit circle. More precisely, for \(0< \alpha <1\) he considers \(A\) consisting of the zeros of the polynomial \((z+ \alpha)^{2n}- (1+\alpha z)^{2n}\) and \(B\) consisting of the zeros of \((z+ \alpha)^n +(1+ \alpha z)^n\) and an additional arbitrary point \(\zeta\) or two additional points \(\zeta\) and 1.
scientific article; zbMATH DE number 1929355