Abstract. The aim of this paper is to generalize the results of [2]. We determine and prove unicity of polynomial $R(x)$ of the lowest possible degree, which has the interpolation properties $R(x_k)=y_k$ ($k=1,\ldots,n$) and $R'(x^*_k)=y_k^\prime $ ($k=1,\ldots,n$), where $x_k$'s are any distinct real nodal points generating the polynomial $w(x)$ and $x_k^*$'s are the roots of $aw(x)+bw'(x)$, where $a$, $b$ are any real numbers, $a\not =0$, $b\not =0$. The case $b=0$ is the Hermite--Fejér interpolation and the case $a=0$ is the Pál interpolation.

AMS Subject Classification (1991): 41A05

Received December 23, 1993; in revised form March 24, 1994. (Registered under 5642/2009.)