Quasi-L p norm orthogonal Galerkin expansions in sums of Jacobi polynomials

Abstract

In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight function, w(x) = (1 − x)α(1 + x)β. However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials, whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only. That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties. Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized in an Lp(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most of these results in specific cases and certain of the results in the general case. However a proof that such expansions can satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.