A continuous spherical wavelet transform for C(Sn)

Wavelet methods in partial differential equations on spheres

“…Theories of continuous spherical wavelets have been developed in the last decades, simultaneously to theories of wavelets over Euclidean space. Iglewska-Nowak has shown in [23,Section 5] that there exist only two essentially different continuous wavelet transforms for spherical signals, namely that based on group theory [2,3] and that derived from approximate identities [10,12,22,23]. In the present paper we show that the latter one can be efficiently applied to solving partial differential equations on the sphere.…”

“…In the first case, the constraints on a function to be a wavelet are quite restrictive, but the inverse transform is given directly by an integral. This is the more popular version of the wavelet transform, developed starting from the 1990s by Freeden et al [12,13,14] and Bernstein et al [5,10], as well as by Iglewska-Nowak in the recent years [22,23,24,25].…”

“…The following definition comes from [22], it is an improved version of the one used in [23], adapted to the case of zonal wavelets and with the most popular weight function…”

“…The wavelet transform is invertible by an integral. Theorem 3.2 in [22] reduced to the case of zonal functions states the following.…”

“…Note that the wavelet transform definition from [22, formula (4)] should be corrected by the factor 1 Σn such that it coincides with the one used in the present paper. The article [22] is based on [23], where the wavelet transform is defined in the same way as in formula (11). In the proof of [22,Theorem 3.2] the reproducing property of the kernels λ+l λ C λ l is used (falsely) without the factor 1 Σn .…”