# Colloquium: Windowed Fourier approximation of analytic functions on the sphere.

### Event Description:

Title: Windowed Fourier approximation of analytic functions on the sphere.

Abstract:

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.