Complementary Lidstone Interpolation and Boundary Value Problems

We shall introduce and construct explicitly the complementary Lidstone interpolating polynomial [inline-graphic not available: see fulltext] of degree [inline-graphic not available: see fulltext], which involves interpolating data at the odd-order derivatives. For [inline-graphic not available: see fulltext] we will provide explicit representation of the error function, best possible error inequalities, best possible criterion for the convergence of complementary Lidstone series, and a quadrature formula with best possible error bound. Then, these results will be used to establish existence and uniqueness criteria, and the convergence of Picard's, approximate Picard's, quasilinearization, and approximate quasilinearization iterative methods for the complementary Lidstone boundary value problems which consist of a [inline-graphic not available: see fulltext]th order differential equation and the complementary Lidstone boundary conditions.