Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

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 ((logn) m ). We compare the obtained Lebesgue constant bound with other well known bounds for Lebesgue constants using different set of points.

Fractional Fokker-Planck Equation

“…It is also optimal in the sense of the Lebesgue measure achieving an optimal value less than Chebyshev approximations [23]. We also note that this behavior is found also in bi-variate approximations based on Poly-Sinc methods [24][25][26].…”

On bivariate poly-sinc collocation applied to patching domain decomposition

“…In [19] it was proved that such interpolation points deliver an accuracy similar to the classical Sinc approximation. The stability of Lagrange approximation at Sinc points is studied in [3,5]. The sequence of Sinc points is generated using a conformal map that redistributes the infinite equidistant points on the real line to a finite interval.…”

“…where, C 1 , C 2 and C 3 are three positive constants, independent of N . For the stability of such approximation, an upper bound of Lebesgue constant has been discussed in [5]. It was shown that Lebesgue's constant for (7) is following the logarithmic relation,…”

“…The exponential conver-gence to the solution is also observed and proved for elliptic partial differential equations (PDE) [4]. The accuracy and stability of Poly-Sinc approximations are discussed in [3,5]. Standard domains in numerical approximation are typically rectangular.…”

“…Standard domains in numerical approximation are typically rectangular. For Poly-Sinc methods rectangular domains ere used to solve Laplace and Poisson's equation for Dirichlet boundaries [3,5]. Unfortunately applications in engineering and sciences require more complicated domains for computations.…”