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 L p(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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Askey, R.: Orthogonal Polynomials and Special Functions. SIAM, Philadelphia (1975)
Andrews, G.E., Burge, W.H.: Determinant identities. Pac. J. Math. 158, 1–14 (1993)
Boyd, J.P.: Chebyshev and Fourier Spectral Methods. Dover, New York (2001)
Doha, E.H., Bhrawy, A.H.: A Jacobi spectral Galerkin method for the integrated forms of fourth-order elliptic differential equations. Numer. Methods Partial Differ. Equ. 25(3) (2009)
Golub, G.H., Van Loan, C.F.: Matrix Computations, third edn. Johns Hopkins, Baltimore (1996)
Gosper, R.W., Jr.: Decision procedure for indefinite hypergeometric summation. Proc. Natl. Acad. Sci. USA 75, 40–42 (1977)
Ierley, G.R.: A class of sparse spectral operators for inversion of powers of the Laplacian in N dimensions. J. Sci. Comp. 12, 57–73 (1997)
Ierley, G.R., Kerswell, R.R., Plasting, S.C.: Infinite Prandtl number convection. Part 2: a singular limit of upper bound theory. J. Fluid Mech. 560, 159–227 (2005)
Ismail, M.: Classical and quantum orthogonal polynomials in one variable. In: Encyclopedia of Mathematics and its Applications, vol. 98. CUP (2005)
Koekoek, R., Swarttouw, R.F.: The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue. Delft University of Technology, Faculty of Information Technology and Systems, Department of Technical Mathematics and Informatics, Report no. 98-17 (1998)
Livermore, P.: Galerkin orthogonal polynomials. J. Comp. Phys. (2009, in press)
Livermore, P.: A compendium of Galerkin orthogonal polynomials. Tech. Rep., Scripps Institution of Oceanography, UC San Diego. Available at http://escholarship.org/uc/item/9vk1c6cm
Luke, Y.: The Special Functions and Their Approximations, vol. 1, first edn. Academic, London (1969)
Marcellan, F., Ronveaux, A.: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math., N.S. 1(4), 451–464 (1990)
Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. Chapman & Hall, London (2003)
Szegö, G.: Orthogonal Polynomials, fourth edn. American Mathematical Society, Providence (1975)
Weber, M., Erdelyi, A.: On the finite difference analogue of Rodrigues’ formula. Am. Math. Mon. 59, 163–168 (1952)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Livermore, P.W., Ierley, G.R. Quasi-L p norm orthogonal Galerkin expansions in sums of Jacobi polynomials. Numer Algor 54, 533–569 (2010). https://doi.org/10.1007/s11075-009-9353-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-009-9353-5