Princeton University

Princeton NJ 08544, USA

Institut de Physique du Globe de Paris

94701 St. Maur-des-Fossés, France

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We pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere, to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

- Figure 01
Sketch illustrating the **geometry**of the spherical concentration problem.

- Figure 02
**Spectral eigensolutions**to the problem of concentrating a spacelimited function within a spectral band.

- Figure 03
**Spatial eigensolutions**to the problem of concentrating a bandlimited spherical harmonic expansion to a circular polar cap.

- Figure 04
**Eigenvalue spectra**of the concentration problem to a circular polar cap, with partial Shannon numbers.*(not included in paper)*

- Figure 05
**Ranked eigenvalues**of the concentration to circular polar caps for a fixed bandwidth. - Figure 06
Cumulative space-domain **energy**of the tapers for the spherical cap problem. - Figure 07
Ranked **Grünbaum eigenvalues**for L=18 and different concentration colatitudes.*(not included in paper)* - Figure 08
**Grünbaum eigenfunctions**for high-degree spherical caps.*(not included in paper)* - Figure 09
First and last four of **Grünbaum eigenfunctions**of the concentration with L=18 and radius 40°. - Figure 10
Eigenfunctions of the concentration to a **circular polar cap**of radius 40° for a fixed bandwidth of L= 18 in decreasing order of concentration. - Figure 11
Eigenvalue structure for the problems of **localizing to different continents**, for different values of the bandwidth L. - Figure 12
Concentrating **Australia**with bandlimited spherical harmonics (L=18). - Figure 13
Concentrating **North America**with bandlimited spherical harmonics (L=18)*(not included in paper)*. - Figure 14
Concentrating **Africa**with bandlimited spherical harmonics (L=18). - Figure 15
Progressive buildup of energy by adding the square of the **continental tapers**weighted by their eigenvalue. - Figure 16
Comparison of several exact **scaled kernels**(grey) with their asymptotic approximation (black).*(not included in paper)* - Figure 17
Approximate number of **significant eigenvalues**given by the exact expression versus the asymptotic results*(not included in paper)*. - Figure 18
Comparison of **scaled spatial eigenfunctions***(not included in paper)*. - Figure 19
Comparison of **scaled spectral eigenfunctions***(not included in paper)*.

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