Title:  Approximations by Jacobi and Generalized Jacobi Polynomials:
Analysis, Algorithms and Applications

 Abstract:
    
      In this talk, we will present a set of optimal Jacobi and
 generalized Jacobi approximation results in non-uniformly weighted
 Sobolev spaces. These results play fundamental roles in the design
 and analysis of spectral-Galerkin methods for a class of partial
 differential equations. In particular, we will illustrate the
 applications of such approximations to the Helmholtz equation (in
 wave scattering), and to high-order differential equations (such as
 KdV-type equations).