Wavelets on the Sphere: Implementation and Approximations

We continue the analysis of the continuous wavelet transform on the 2-sphere, introduced in a previous paper. After a brief review of the transform, we define and discuss the notion of directional spherical wavelet, i.e., wavelets on the sphere that are sensitive to directions. Then we present a calculation method for data given on a regular spherical grid G. This technique, which uses the FFT, is based on the invariance of G under discrete rotations around the z axis preserving the φ sampling. Next, a numerical criterion is given for controlling the scale interval where the spherical wavelet transform makes sense, and examples are given, both academic and realistic. In a second part, we establish conditions under which the reconstruction formula holds in strong L sense, for 1 ≤ p < ∞. This opens the door to techniques for approximating functions on the sphere, by use of an approximate identity, obtained by a suitable dilation of the mother wavelet.