Thoughts on harmonic analysis on the sphere: spherical wavelet frames and kernels

In the central spirit of harmonic analysis lies the concept of effectively decomposing, analyzing and representing functions or functionals. It has lead to the flourish of Fourier analysis and its modern descendants such as wavelets and its siblings. Especially, the construction of spherical wavelets and its related theory is at junior age. This dissertation gives a brief summary of existing results in this field on the one hand, creates spherical alpha-wavelets and further develops spherical kernel theory on the other hand. Among various strategies, two types of spherical wavelets are emphasized, one constructed in the frequency domain, the other generated through stereographic projection. In the former one I discuss localized tight frame design and its directional extension. In the latter one a new anisotropic dilation is defined, and a representation system generated by it consists of the so called spherical \alpha-wavelets/shearlets. Summability properties of those wavelets/shearlets are well established once they are restricted to certain subspaces of square-integrable functions newly defined in this dissertation, including the so called hollow pole functions. Kernels, though deeply rooted in classical theory, can find its variation and application in the frame theory. Indeed, frame kernel, a concept which is proposed in this dissertation, is an equivalent formulation to the frame itself. Besides, there exist a variety of kernels which exhibit their own special properties. For zonal kernels, I give its necessary and sufficient conditions to approximate square integrable functions on the sphere. Multiscale kernel, a recently appeared concept, will meet its spherical version here and it turns out to have reproducing property for certain Hilbert space of spherical functions. One of the climaxes in this work is the invention of two novel frames, based on the two spherical wavelets constructions. In the zonal kernel approach I give frame properties inside the multiresolution structure; while for alpha-wavelets, I prove that under certain conditions they form tight frames in continuous and discrete setting respectively, following from which are reproducing formulae that enable us to reconstruct or approximate numerically an integrable function or solutions of PDEs. Based on the obtained frames a spherical Galerkin scheme is proposed afterwards. At the end of this dissertation I give an inner product formula with respect to a recently emerged surface-value dependent inner product space on a triangular mesh and prove its equivalence to the combinatoric inner product.

Author dc:creator

• Sun, Yizhi

Advisor dc:contributor.advisor

• Schneider, Reinhold

Licence dc:rights.uri

• https://creativecommons.org/licenses/by-nc-nd/4.0/

Language dc:language.iso

en

Identifier URI

http://dx.doi.org/10.14279/depositonce-9808

OAI identifier oai:identifier

oai:depositonce.tu-berlin.de:11303/10915