An alternative to Slepian functions on the unit sphere – A space–frequency analysis based on localized spherical polynomials

Abstract

In this article, we present a space-frequency theory for spherical harmonics based on the spectral decomposition of a particular space-frequency operator. The presented theory is closely linked to the theory of ultraspherical polynomials on the one hand, and to the theory of Slepian functions on the 2-sphere on the other. Results from both theories are used to prove localization and approximation properties of the new band-limited yet space-localized basis. Moreover, particular weak limits related to the structure of the spherical harmonics provide information on the proportion of basis functions needed to approximate localized functions. Finally, a scheme for the fast computation of the coefficients in the new localized basis is provided.

Shapes of Uncertainty in Spectral Graph Theory

“…This simplifies the methods derived in [1] for the calculation of the convex uncertainty curve. • In a Landau-Pollak-Slepian type space-frequency analysis, it is possible to formulate error estimates for the approximation of space-frequency localized signals [9,10,20]. In our general framework on graphs, these estimates will be derived in Section 6.…”

“…x 2 . Similar distance-projection filters were used in a continuous setup for orthogonal expansions on the interval [−1, 1] [8, 9, 18] and on the unit sphere [10].…”

A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

“…Simons et al [81] gave an up-to-date review of time-frequency and time-scale concentration problems on a sphere. We also refer to the recent works [21,109] along this line. SenGupta et al [75] considered the concentration problem over disjoint frequency intervals.…”

Weak limits for weighted means of orthogonal polynomials

“…In this paper we want to derive and investigate analogs of the weak limits (1), ( 2) and (3) for weighted means of differing orthogonality measures. The motivation to study such mean weak limits originates in two works [5] and [6] in which a Landau-Pollak-Slepian type spacefrequency analysis was studied for spaces of orthogonal polynomials. In the one-dimensional case given in [5], the roots of orthogonal polynomials were used to describe the spatial position of localized basis functions.…”

“…The corresponding asymptotic distribution of the roots is given by the arcsine distribution (3). In the case of the unit sphere a similar description was derived in [6]. This description however included ultraspherical and co-recursive ultraspherical polynomials with a differing parameter.…”