## Polynomial approximations to functions of operators.

##### Abstract

To solve the linear equation Ax = f, where f is an element of Hilbert space H and A
is a positive definite operator such that the spectrum (T (A) ( [m,M] , we approximate
-1
the inverse operator A by an operator V which is a polynomial in A. Using the
spectral theory of bounded normal operators the problem is reduced to that of
approximating a function of the real variable by polynomials of best uniform
approximation. We apply two different techniques of evaluating
A-1 so that the
operator V is chosen either as a polynomial P (A) when P (A) approximates the
n n
function 1/A on the interval [m,M] or a polynomial Qn (A) when 1 - A Qn
(A)
approximates the function zero on [m,M]. The polynomials Pn (A) and Qn (A)
satisfy three point recurrence relations, thus the approximate solution vectors P (A)f
n
and Q (A)f can be evaluated iteratively. We compare the procedures involving
n
Pn (A)f and Qn (A)f by solving matrix vector systems where A is positive definite.
We also show that the technique can be applied to an operator which is not selfadjoint,
but close, in the sense of operator norm, to a selfadjoint operator. The iterative
techniques we develop are used to solve linear systems arising from the discretization of
Freedholm integral equations of the second kind. Both smooth and weakly singular
kernels are considered. We show that earlier work done on the approximation of linear
functionals < x,g > , where 9 EH, involve a zero order approximation to the inverse
operator and are thus special cases of a general result involving an approximation of
arbitrary degree to A -1 .