Bivariate Lagrange interpolation at the Padua points: The generating curve approach

Abstract

We give a simple, geometric and explicit construction of bivariate interpolation at certain points in a square (called Padua points), giving compact formulas for their fundamental Lagrange polynomials. We show that the associated norms of the interpolation operator, i.e., the Lebesgue constants, have minimal order of growth of O((log n) 2 ). To the best of our knowledge this is the first complete, explicit example of near optimal bivariate interpolation points.

Multivariate Lagrange Interpolation at Sinc Points Error Estimation and Lebesgue Constant

“…To this end, we first introduce the bivariate case. In this case, we compare with bivariate Lagrange approximations using different sets of points, like Chebyshev, Padua, Xu, Lissajous points discussed in (Rivilin, 1974), (Brutman, 1997), (Bos and Xu et al 2006), (Caliari et al, 2005) and, (Erb et al, 2015). We then compare the Lagrange approximations at Sinc data with Sinc approximation.…”

“…For example, using Padua and Xu points as interpolation points can yield a rate of O(log(n)) 2 , see , (Bos and Xu et al, 2006), and (Vecchia et al, 2009). Also using Lissajous points can follow the same asymptotic behavior, see (Erb et al, 2015).…”

Weakly Admissible Meshes and Discrete Extremal Sets

“…Moreover, using property P2 and P7, WAMs for the triangle and for linear trapezoids, again with approximately 2n 2 points and C(A n ) = O(log 2 n), have been obtained simply by mapping the so-called Padua points of degree 2n from the square by standard quadratic transformations (the first known optimal points for bivariate polynomial interpolation, with a Lebesgue constant growing like log-squared of the degree, cf. [3]). …”

Padua2DM: fast interpolation and cubature at the Padua points in Matlab/Octave

“…, n(n + 1) where γ(t) is their "generating curve" (cf. [1]) γ(t) = (− cos((n + 1)t), − cos(nt)), t ∈ [0, π]…”

“…Such a cubature formula is derived from quadrature along the generating curve and is the key to obtaining the fundamental Lagrange polynomials (2); cf. [1].…”