In 1974 L. G. Pál introduced a modification of the Hermite-Fejér interpolation. This problem is a special case of Birkhoff interpolation that was discovered in 1906 by G. D. Birkhoff. In the first chapter we introduce a generalization of the Pál interpolation process. Our interpolation method has the advantage of existence for every n and uniqueness without any additional nodal points x0, x0*. In Sections 2 and 3 we apply the result of Section 1 for special cases to prove convergence theorems in weighted sup-norm. In the second chapter we consider the Lagrange and Hermite interpolation processes based on the roots of orthogonal polynomials. We prove convergence theorems in weighted L p -norm.