Computability in Europe 2008
Logic and Theory of Algorithms

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Regular Talk:
Algorithms for Analytic Functions and Applications to Toeplitz Operators

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Author(s): Edwin Beggs and Annelies Gerber
Slot: Mon, 12:00-12:20, Vice-Rector (col. 4)

Abstract

In this paper we present two algorithms for analytic functions. They are based
on a quadtree datatype and we use methods from object oriented programming to
describe them. The first algorithm decides at each level n whether a given
square of length $2^{-n}$ lies further than a minimal distance away from the
graph of a function f(z) and then computes the winding number for the centre of
this square with respect to f(z). The second algorithm computes zeros of an
analytic function f(z). Many algorithms for computing zeros of analytic
functions exist already. Our algorithm explicitly deals with the problem of when
zeros occur on the boundaries of squares and when the domain of the function is
potentially complicated. We propose a method of perturbing the relevant edges of
such squares to guarantee the method working. Then, we briefly explain how we
can apply these algorithms to particular linear operators on Hilbert spaces
called Toeplitz operators. The first algorithm allows us to draw the spectrum of
such a Toeplitz operator T(a). The second algorithm is used to determine the
spectrum of a perturbed Toeplitz operator.

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