Spherical harmonic analysis and model-limited extrapolation on the sphere: Integral equation formulation

The classical problem of extrapolation of a bandlimited signal from limited time domain data is revisited for signals defined on the sphere. That is, given limited or incomplete measurements of a low pass signal on the unit sphere, find the unique extrapolation to the complete unit sphere. Signals defined on the unit sphere arise in a number of applications, such as beam-patterns in azimuth and elevation and head related transfer functions. Our investigations explore the role of integral equation operators in characterizing the extrapolation problem which leads to an iterative algorithm analogous to that obtained in the time-frequency case.