Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

Abstract: We construct an adaptive estimator of a density function on $d$ dimensional unit sphere $S^d$ ($d \geq 2 $), using a new type of spherical frames. The frames, or as we call them, stereografic wavelets are obtained by transforming a wavelet system, namely Daubechies, using some stereographic operators. We prove that our estimator achieves an optimal rate of convergence on some Besov type class of functions by adapting to unknown smoothness. Our new construction of stereografic wavelet system gives us a multiresolution approximation of $L^2(S^d)$ which can be used in many approximation and estimation problems. In this paper we also demonstrate how to implement the density estimator in $S^2$ and we present a finite sample behavior of that estimator in a numerical experiment.