An interpolation process on the roots of the integrated Legendre polynomials (Q802089)

Let \(\{x_{k,n}\}\) be the n zeros of \(\pi_ n(x)=(1-x^ 2)P'_{n-1}(x)\), where \(P_{n-1}(x)\) is the (n-1)th Legendre polynomial and let \(\{x^*_{k,n}\}\) be n-1 zeros of \(\pi'_ n(x)\). Obviously \(-1= x_{n,n} < x^*_{n-1,n} < x_{n-1,n} <...< x_{2,n} < x^*_{1,n} < x_{1n} =1\). The author considers the problem of finding a polynomial \(R_ n\) of degree \(2n-2\) at most such that \(R_ n(x^*_{i,n})=y_{i,n}\) \((i=1,2,...,n-1)\); \(R'_ n(x_{i,n})=y'_{i,n}\) \((i=1,2,...,n)\), where \(y_{i,n}\) and \(y'_{i,n}\) are arbitrarily given real numbers. This problem is uniquely solvable only if n is even. For n odd there is a polynomial \(c\pi'_ n(x)[P_{n-1}(x)-2]\) with c an arbitrary real number which satisfies the requirements of the problem. As to the convergence of the sequence of interpolation polynomials \(R_ n(f,x)\) \((n=2,4,6,...)\), he has shown that if \(f\) is continuously differentiable on \([-1,1]\) then \(| R_ n(f,x)-f(x)| =O(n^{1/2}\omega (f';1/n))\), \(-1\leq x\leq 1\), where \(\omega(f',\delta)\) is the modulus of continuity of \(f'\) and 0 does not depend on \(x\).