IJPAM: Volume 1, No. 1 (2002)
APPROXIMATION METHODS FOR
Applied Sciences Dept.
College of Technological Studies, KUWAIT
Dept. of Mathematics,
Imperial College, London, U.K.
Abstract.We introduce sequences of linear functionals uniquely associated with each of several methods for the approximate solution of differential equations and discuss the classification of the latter in terms of these functionals. We use Davis's [#!dav!#] notion of permanence to discuss the structure of these methods further. We show that the Tau method, formulated in terms of sequences of canonical polynomials (see Ortiz [#!ort69!#]), is a permanent technique which makes no direct reference to a Hilbert structure. We also show that, for a given operator, the linear functionals associated with it are biorthogonal to the canonical polynomials. Furthermore, we make precise here a duality which exists between the Tau method and collocation, using the linear functionals associated with each of these methods. We relate them to spectral or Fourier expansion techniques.
Received: January 6, 2002
AMS Subject Classification: 65L60, 65J10
Key Words and Phrases: interpolation, collocation, Tau method, canonical polynomials, orthogonal polynomials
Source: International Journal of Pure and Applied Mathematics