Colloquium: Windowed Fourier approximation of analytic functions on the sphere.
Title: Windowed Fourier approximation of analytic functions on the sphere.
Windowed Fourier transforms have long been used to study local
features of signals and are often called short-time Fourier
transforms. They are commonly used to determine frequency and
phase content of local sections of a signal as it changes over
time. They are, however, rarely used in the solution and analysis
of differential equations because regions near the boundaries of
the domain are difficult to handle. Spherical geometries and more
general manifolds, on the other hand, are boundary free, and
windowed Fourier transforms provide an excellent framework for
approximation and numerical solution of PDEs on these surfaces.
In this talk a spectral method based on windowed Fourier
approximations for computations on the sphere is presented. It
relies on domain decomposition, such as the cubed sphere, and is
suitable for adaptive and parallel implementation. One of the
advantages of this approach is that computations can be carried
out using fast Fourier transforms on a nearly uniform grid.
Approximations are obtained on overlapping domains and a global
solution is obtained using partition of unity. A convergence
analysis for analytic functions will be presented.
Contact Name: Stephen Lau
Contact Email: email@example.com