Polynomial approximation on the sphere using scattered data

The paper tackles the problem of approximately reconstructing a real function defined on the surface of the unit sphere in the Euclidean q-dimensional space, with q>1, starting from function's samples at scattered sites. Two new operators are introduced for continuous and discrete approximation at scattered sites. Moreover precise error estimates as well as Marcinkiewicz-Zygmund inequalities are derived in every Lp space, giving concrete bounds for all the involved constants.