Abstract. We introduce a family of generalized Jacobi
polynomials/functions with indexes α, β ∈ R which are mutually
orthogonal with respect to the corresponding Jacobi weights and which
inherit selected important properties of the classical Jacobi
polynomials. We establish their basic approximation properties in
suitably weighted Sobolev spaces. As an example of their applications,
we show that the generalized Jacobi polynomials/functions, with
indexes corresponding to the number of homogeneous boundary conditions
in a given partial differential equation, are the natural basis
functions for the spectral approximation of this partial differential
equation. Moreover, the use of generalized Jacobi
polynomials/functions leads to much simplified analysis, more precise
error estimates and well conditioned algorithms.