On some modified \((0,\dots, m-2,m)\) interpolation (Q2725251)

Let \(x_1,\dots,x_{n+2}\) be the zeros of \((1-x)^2P_n(x)\), where \(P_n\) denotes the Legendre polynomial of degree \(n\). The authors consider the following modified \((0,1,\dots,m-2,m)\) interpolation problem: Find polynomials \(R_n\) of degree \(\leq m(n+2)-5\), \(m>2\), satisfying the conditions NEWLINE\[NEWLINE\begin{aligned} R_n^{(p)} (x_i)=y_i^{(p)}, \quad & i=1,2,\dots, n+2;\;p=0,1,\dots, m-3\\ R_n^{(p)} (x_i)=y_i^{(p)}, \quad & i=2,3,\dots, n+1;\;p=m-2,m, \end{aligned}NEWLINE\]NEWLINE where the \(y_i^{(p)}\) are pre-assigned real numbers. They show that the problem has a unique solution for any set of values \(y_i^{(p)}\) if and only if \(n\) and \(m\) are even. A similar \((0,2)\) interpolation problem is also studied.