Orthogonal non-bandlimited wavelets on the sphere
Willi Freeden, Volker Michel · 2000
This paper introduces orthogonal non--bandlimited wavelets on the sphere with respect to a certain Sobolev space topology. The construction of those kernels is based on a clustering of the index set N = f(n# k) 2 0 \\Theta Zj;n k ng associated to the system of spherical harmonics fY n#k g (n#k)2N . The wavelets presented here form reproducing kernels of the spans of the clustered harmonics. More explicitly, the horizontal partition Mn = f(n# k) 2 Ng, n 2 N yields the usual Shannon wavelets, which are bandlimited, whereas non-bandlimited kernels can be obtained from a vertical clustering B k = f(n# k)# (n# ;k) 2 Ng, k 2 N 0 . For this case a particular kernel is investigated in detail, and a wavelet representation is derived explicitly. AMS classification: Primary: 42C40, 65T60. Secondary: 86-08. Key Words: spherical approximation, orthogonal bandlimited and non--bandlimited wavelets, explicit representations. 1