WEIGHTED HAAR WAVELETS ON THE SPHERE

Starting from the one-dimensional Haar wavelets on the interval [0,1], we construct spherical Haar wavelets which are orthogonal with respect to a given scalar product. This scalar product induces a norm which is equivalent to the usual ‖ · ‖₂ norm of L²(𝕊²). Thus, the Riesz stability in L²(𝕊²) is assured and we can use the algorithms of decomposition and reconstruction from the Haar wavelets in 2D. Another advantage is that we avoid the problems around the poles, which occur in other approaches. As example, we decompose a data set, showing the graphs of the approximations and details and thus the capability to detect the singularities (contours). The method described here can be also used for constructing spherical wavelets starting from wavelets on an interval.