Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations

Jacobi approximations in certain Hilbert spaces are investigated. Several w eighted inverse inequalities and Poincare inequalities are obtained. Some a pproximation results are given. Singular differential equations are approxi mated by using Jacobi polynomials. This method keeps the spectral accuracy. Some linear problems and a nonlinear logistic equation are considered. The stabilities and the convergences of proposed schemes are proved strictly. The main idea and techniques used in this paper are also applicable to othe r singular problems in multiple-dimensional spaces. (C) 2000 Academic Press .