Modified Taylor reproducing formulas and h-p clouds

We study two different approximations of a multivariate function $f$ by operators of the form $\sum_{i=1}^{N}\mathcal{T}_{r}[f,x_{i}](x)\,\mathcal{W} _{i}(x)$, where $\{\mathcal{W}_{i}\}$ is an $m$-reproducing partition of unity and $\mathcal{T}_{r}[f,x_{i}](x)$ are modified Taylor polynomials of degree $r$ expanded at $x_{i}$. The first approximation was introduced by Xuli (2003) in the univariate case and generalized for convex domains by Guessab et al. (2005). The second one was introduced by Duarte (1995) and proved in the univariate case. In this paper, we first relax the Guessab's convexity assumption and we prove Duarte's reproduction formula in the multivariate case. Then, we introduce two related reproducing quasi-interpolation operators in Sobolev spaces. A weighted error estimate and Jackson's type inequalities for h-p cloud function spaces are obtained. Last, numerical examples are analyzed to show the approximative power of the method.