Kernels of Spherical Harmonics and Spherical Frames

Our concern is with the construction of a frame in L 2 (S) consisting of smooth functions based on kernels of spherical harmonics. The corresponding decomposition and reconstruction algorithms utilize discrete spherical Fourier transforms. Numerical examples connrm the theoretical expectations. x1. Introduction Traditionally, wavelets were tailored to problems on the Euclidean space IR d. However , in most applications one has to analyze functions deened on compact domains. In particular, in geophysics wavelets on the unit sphere S of IR 3 are of interest. There exist diierent approaches to the constructions of spherical wavelets. Having spherical coordinates in mind, the idea of using tensor{products of periodic wavelets and wavelets on the interval was suggested in 7]. Applying tensor{products of periodic exponential spline{wavelets and spline{wavelets on the interval, wavelets on S were constructed in 3]. Unfortunately, tensor{product wavelets can possess singularities at the poles of S. To avoid these singularities, the coeecients of scaling functions and wavelets have to satisfy a linear system on each level, whose dimension grows very fast (see 3]). A tensor{product method with a completely diierent kind of wavelets, namely trigonometric wavelets 2] and polynomial wavelets 11], was considered in 10]. But these wavelets cannot be used for the detection of singularities of a given function at the poles of S.