Quasi-interpolation in Riemannian manifolds (Q2841057)

Quasi-interpolation is a highly relevant and useful approach to functional approximation within the general context of approximation theory. A quasi-interpolation operator usually maps functions, employing their values evaluated on grids, scaled grids or even scattered or quasi-uniformly data, to linear combinations of translates and scales of (often) a single basis function, using the aforementioned function values as coefficients of these linear combinations. In the interpolation (rather than in the quasi-interpolation) case, these basis functions would be Lagrange functions, but here they have to satisfy weaker conditions such as asymptotic decay and certain recovery properties to replace the Lagrangian conditions. The latter recovery properties are designed so that the approximation operator in use provides some accuracy for instance when smooth approximands are approximated. This general idea is taken in this article and generalised to Riemannian manifolds whereas in the classical case, only Euclidean spaces are used. Convergence theorems are established in this new context as well as numerical experiments and results are given.