On some open problems of P. Turán concerning Birkhoff interpolation

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

$${\pi _n}(x) = {c_n}\int_{ - 1}^x {{P_{n - 1}}(t)\,dt}$$

, ${P_n}(x)$ being Legendre polynomial of degree $n$.