Regularity of (0,1,... r-2,r) and (0,1,...,r-2,r) Interpolations on Some Sets of the Unit Circle

The purpose of thes paper is to show regularity of (0,1,... r-2,r) and (0,1,... r-2,r)^* interpolations on the sets obtained by projecting vertically the zeros of 

\((1-x^2)P_n^{(\alpha , \beta)} \ \ \ \ \ \ (-1<\alpha , \beta <= \frac {1}{2})\\ (1-x)P_n^{(\alpha , \beta)} \ \ \ \ \ \ (-1<\alpha<= \frac {1}{2} ,-1< \beta <= \frac {1}{2})\)

and

\((1+x)P_n^{(\alpha , \beta)} \ \ \ \ \ \ (-1<\alpha<=\frac {1}{2} , -1 <\beta <= \frac {1}{2})\\ \)

 respectively onto the unit circle, where \(P_n^{ ({\alpha , \beta})}(x)\) stand for the th Jacobi polynomial.