Interpolatory Wavelets on the Sphere

In this paper, we construct an interpolatory wavelet basis on the unit sphere S l R 3. Using spherical coordinates, we apply the tensor product of interpolatory trigonometric and algebraic polynomial wavelets. The described decomposition and reconstruction algorithms work in the frequency domain. x0. Introduction The wavelet theory is closely related to shift{invariant subspaces of L 2 (l R d). Therefore, wavelets are naturally adapted to problems on the whole space l R d. Very often, one has to deal with functions deened on a bounded domain such that the construction of wavelets on a domain is desirable. This problem is solved for the torus T := l R=2Z Z 5], and for compact intervals 1, 4]. In the sequel, we will apply 2{periodic trigonometric polynomial wavelets 6] and algebraic polynomial wavelets on I := ?1; 1] 3, 9] in order to construct interpolatory wavelets on the unit sphere S l R 3. Other approachs to wavelets on the sphere can be found in 2, 7]. Using modiied spherical coordinates x := (x 1 ; x 2) 2 H := T I, we can identify the sphere S with H by the mapping :