Generalized Hyperinterpolation on the Sphere and the Newman—Shapiro Operators

Abstract

Hyperinterpolation on the sphere, as introduced by Sloan in 1995, is a constructive approximation method which is favorable in comparison with interpolation, but still lacking in pointwise convergence in the uniform norm. For this reason we combine the idea of hyperinterpolation and of summation in a concept of generalized hyperinterpolation. This is no longer projectory, but convergent if a matrix method A is used which satisfies some assumptions. Especially we study A partial sums which are defined by some singular integral used by Newman and Shapiro in 1964 to derive a Jackson-type inequality on the sphere. We could prove in 1999 that this inequality is realized even by the corresponding discrete operators, which are generalized hyperinterpolation operators. In view of this result the Newman—Shapiro operators themselves gain new attention. Actually, in their case, A furnishes second-order approximation, which is best possible for positive operators. As an application we discuss a method for tomography, which reconstructs smooth and nonsmooth components at their adequate rate of convergence. However, it is an open question how second-order results can be obtained in the discrete case, this means in generalized hyperinterpolation itself, if results of this kind are possible at all.