Continuous wavelet transforms on n-dimensional spheres

Abstract

In this paper, we are concerned with n-dimensional spherical wavelets derived from the theory of approximate identities. For nonzonal bilinear wavelets introduced by Ebert et al. in 2009 we prove isometry and Euclidean limit property. Further, we develop a theory of linear wavelets. In the end, we discuss the relationship to other wavelet constructions.

On the uncertainty product of spherical functions

“…This concept -generalized to the n-dimensional case -was also studied by the author of the present paper. I showed in [19] that directional derivatives of some zonal wavelets satisfy a slightly modified definition of wavelets derived from approximate identities (see [11,10,9,1,8,3,2] for the origins of this concept and [18] for a comprehensive survey). In [20] I proposed further relaxation on the constraints on wavelets derived from approximate identities and showed that directional derivatives of a wide class of functions are wavelets according to the new definition.…”

A continuous spherical wavelet transform for C(Sn)

“…This section is devoted to the bilinear wavelet transform as defined in [19,Section 3]. The goal is to relax the constraints on a function family to be a wavelet, more precisely, to abandon the condition…”

“…[1,2,3,4,5,6,7,8,9,11,12,13,14,15,17,18,19,20,22]. I have shown in [19,Section 5] that there are only two essentially different continuous wavelet transforms for spherical signals, namely that based on group theory [2,3] and that derived from approximate identities [10,7,19]. In both cases, square integrable signals are considered, and the inverse transform is performed by an integral that converges in L 2 -norm.…”

“…The novelty is that the inverse wavelet transform converges uniformly. The paper is based on [19] and the same notation is used. In Section 2 a new (with respect to that from [19]) definition of a wavelet family is given, and it is shown that the wavelet analysis and the wavelet synthesis can be performed in the usual way.…”

“…The paper is based on [19] and the same notation is used. In Section 2 a new (with respect to that from [19]) definition of a wavelet family is given, and it is shown that the wavelet analysis and the wavelet synthesis can be performed in the usual way. In Section 3 a wavelet family is constructed such that the wavelet transform is invertible in the supremum norm.…”