IOP PUBLISHING INVERSE PROBLEMS Inverse Problems 23 (2007) 2059–2088 doi:10.1088/0266-5611/23/5/015 Grids and transforms for band-limited functions in a disk Gregory Beylkin, Christopher Kurcz and Lucas Monz´ on Department of Applied Mathematics, University of Colorado at Boulder, 526 UCB, Boulder, CO 80309-0526, USA Received 24 April 2007, in final form 7 August 2007 Published 28 August 2007 Online at stacks.iop.org/IP/23/2059 Abstract We develop fast discrete Fourier transforms (and their adjoints) from a square in space to a disk in the Fourier domain. Since our new transforms are not unitary, we develop a fast inversion algorithm and derive corresponding estimates that allow us to avoid iterative methods typically used for inversion. We consider the eigenfunctions of the corresponding band-limiting and space-limiting operator to describe spaces on which these new transforms can be inverted and made useful. In the process, we construct polar grids which provide quadratures and interpolation with controlled accuracy for functions band-limited within a disk. For rapid computation of the involved trigonometric sums we use the unequally spaced fast Fourier transform, thus yielding fast algorithms for all new transforms. We also introduce polar grids motivated by linearized scattering problems which are obtained by discretizing a family of circles. These circles are generated by using a single circle passing through the origin and rotating this circle with the origin as a pivot. For such grids, we provide a fast algorithm for interpolation to a near optimal grid in the disk, yielding an accurate adjoint transform and inversion algorithm. 1. Introduction This paper introduces fast discrete Fourier transforms from a square in the spatial domain to a disk in the Fourier domain. Whereas there are many possible discretizations of the Fourier transform, the new transforms are special in that, under appropriate conditions, we may use the adjoint in lieu of the inverse transform. Unlike the discrete Fourier transform (DFT), these transforms are not unitary and have a numerical null space. For these reasons, we pay special attention to describing spaces on which they may be inverted and made useful. We develop a fast inversion algorithm (and corresponding estimates) that avoids iterative methods typically used for inversion. We note that the speed of all new algorithms is proportional to that of the fast Fourier transform (FFT). In essence, we present an approach for constructing useful analogues of the FFT to and from a polar grid in the Fourier domain. 0266-5611/07/052059+30$30.00 © 2007 IOP Publishing Ltd Printed in the UK 2059