Computing with functions on the disk

This talk addresses some of the challenges associated with numerically representing functions on the unit disk, and offers a new approximation method. Low rank approximations to functions in polar geometries are formed by synthesizing the disk analogue of the double Fourier sphere method with a structure-preserving variant of iterative Gaussian elimination. This adaptive procedure is near-optimal in its sampling strategy, producing approximants that are stable for differentiation, smooth over the origin, and facilitate the use of FFT-based algorithms in both the radial and angular directions. The low rank form of the approximants is especially useful for operations such as integration and differentiation, reducing them to essentially 1D procedures. An optimal Poisson solver is formulated by combining this strategy with Fourier and ultraspherical spectral methods.