Abstract
This paper gives an explicit construction of multivariate Lagrange interpolation at Sinc points. A nested operator formula for Lagrange interpolation over an $m$-dimensional region is introduced. For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior. For the uniform convergence the growth of the associated norms of the interpolation operator, i.e., the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature $O((log n)^m)$. We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.
Highlights
One of the fundamental problems in approximation theory is the Lagrange interpolation problem: given a continuous function f, a set of interpolation points X and a polynomial interpolation space Π, and Pn an element of Π which is equal to f at the interpolation points X
For the nested Lagrange interpolation, a proof of the upper bound of the error is given showing that the error has an exponentially decaying behavior
The following numerical experiments examine the calculations of the error and Lebesgue constant for multivariate Lagrange approximation at Sinc points
Summary
One of the fundamental problems in approximation theory is the Lagrange interpolation problem: given a continuous function f , a set of interpolation points X and a polynomial interpolation space Π, and Pn an element of Π which is equal to f at the interpolation points X. Interpolation using spaced points has been shown to yield undesirable divergent behaviors even for analytical functions as the polynomial degree of interpolation increases. This behavior is called Runge phenomenon, see (Runge, 1901). For the 1D Lagrange interpolation problem, the explicit construction of the interpolating polynomial for certain interpolation points and the derivation of remainder formulas, we refer the reader to the works (Smith, 2006), (Youssef et al, 2016), and (Rivilin, 1974) and references therein This interpretation of Lebesgue constant is not restricted to polynomial approximation but can be used to judge the accuracy and stability of any approximation.
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