Multivariate approximation by a combination of modified Taylor polynomials
Journal of Computational and Applied Mathematics
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Let $\mathfrak{P}$ be a convex polytope in the d-dimensional Euclidean space. We consider an interpolation of a function f at the vertices of $\mathfrak{P}$ and compare it with the interpolation of f and its derivative at a fixed point $y\in\mathfrak{P}.$ The two methods may be seen as multivariate analogues of an interpolation by secants and tangents, respectively. For twice continuously differentiable functions, we establish sharp error estimates with respect to a generalized Lp norm for $1\le p\le\infty$. The case p = 1 is of special interest since it provides analogues of the midpoint rule and the trapezoidal rule for approximate integration over the polytope $\mathfrak{P}.$ In the case where $\mathfrak{P}$ is a simplex and p 1, this investigation covers recent results by S. Waldron [SIAM J. Numer. Anal., 35 (1998), pp. 1191--1200] and by M. Stämpfle [J. Approx. Theory, 103 (2000), pp. 78--90].