On some open problems of P. Turán concerning Birkhoff interpolation

Source abstract

In 1974 P. Turán (see [ 6 ]) raised many interesting open problems in Approximation Theory some of which are on Birkhoff Interpolation. The object of this paper is to answer some of these problems (XXXVI-XXXIX). We obtain some new quadrature formulas where function values and second derivatives are only prescribed on the zeros of $$π n ( x ) = c n ∫ − 1 x P n − 1 ( t ) d t {\pi _n}(x) = {c_n}\int _{ - 1}^x {{P_{n - 1}}(t)\,dt}$$ , P n ( x ) {P_n}(x) being Legendre polynomial of degree n n .