Asymptotic formulae of multivariate Bernstein approximation
Journal of Approximation Theory
Asymptotic expansions for multivariate polynomial approximation
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 vol. II: interpolation and extrapolation
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
Multi-node higher order expansions of a function
Journal of Approximation Theory
Error estimates for modified local Shepard's interpolation formula
Applied Numerical Mathematics
On the approximability and the selection of particle shape functions
Numerische Mathematik
Sharp Error Estimates for Interpolatory Approximation on Convex Polytopes
SIAM Journal on Numerical Analysis
Increasing the polynomial reproduction of a quasi-interpolation operator
Journal of Approximation Theory
Optimal bivariate C1 cubic quasi-interpolation on a type-2 triangulation
Journal of Computational and Applied Mathematics
Error analysis for a non-standard class of differential quasi-interpolants
Mathematics and Computers in Simulation
Enhancing the approximation order of local Shepard operators by Hermite polynomials
Computers & Mathematics with Applications
The finite element method enriched by interpolation covers
Computers and Structures
Construction techniques for multivariate modified quasi-interpolants with high approximation order
Computers & Mathematics with Applications
Hi-index | 7.29 |
We study an approximation of a multivariate function f by an operator of the form Σi=1N˜r|f, xi|(x)φi(x), where φ1,...,φN are certain basis functions and ˜r|f, xi|(x) are modified Taylor polynomials of degree r expanded at xi. The modification is such that the operator has highest degree of algebraic precision. In the univariate case, this operator was investigated by Xuli [Multinode higher order expansions of a function, J. Approx. Theory 124 (2003) 242-253]. Special attention is given to the case where the basis functions are a partition of unity of linear precision. For this setting, we establish two types of sharp error estimates. In the two-dimensional case, we show that this operator gives access to certain classical interpolation operators of the finite element method. In the case where φ1,...,φN are multvariate Bernstein polynomials, we establish an asymptotic representation for the error as N → ∞