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Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets. (English) Zbl 1405.42060

An adaptive estimator of a density function on the \(d\)-dimensional unit sphere \(S^{d}\) is constructed. A new type of spherical frames is used. These frames are obtained by transforming a wavelet system, namely Daubechies, using some stereographic operators. It is proved that this estimator achieves an optimal rate of convergence on some Besov type class of functions by adapting to unknown smoothness. A new construction of the stereographic wavelet system gives a multiresolution approximation of \(L^{2}(S^{d})\) which can be used in many approximation and estimation problems. It is demonstrated how to implement the density estimator in \(S^{2}\). A finite sample behavior of that estimator is presented in a numerical experiment.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A30 Approximation by other special function classes
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