Iterated solutions of linear operator equations with the Tau method

The Tau Method produces polynomial approximations of solutions of differential equations. The purpose of this paper is (i) to extend the recursive formulation of this method to general linear operator equations defined in a separable Hilbert space, and (ii) to develop an iterative refinement procedure which improves on the accuracy of Tau approximations. Applications to Fredholm integral equations demonstrate the effectiveness of this technique. 1. Canonical polynomials of linear operators Let X be a separable Hilbert space having a basis x := {x0, x1, x2, . . . } , and let A(X) denote the space of all linear operators on X . Then, for A ∈ A(X) and for all i ∈N, Axi is a linear combination of the basis elements: Axi = ∑ j aijxj , aij ∈ C. (1) For A ∈ A(X) and x, y ∈ X , we define the equivalence relation x ≡A y iff x− y ∈ kerA . An operator A is said to be banded-from-above if there exists an integer k ∈ N which satisfies the property P (k) defined as: aij = 0 ∀i, j ≥ 0 with j − i ≥ k + 1. Let Ab(X) ⊆ A(X) denote the subspace of all banded-from-above operators on X , and define an integer-valued function on the elements of Ab(X) by h(A) := min{k ∈ N : P (k)} which, following Ortiz [3, 4], is called the height of A . Then, if A ∈ Ab is of height h(A) equation (1) becomes Axi = h+i ∑