Small perturbations of polynomial meshes

Abstract

We show that the property of being a (weakly) admissible mesh for multivariate polynomials is preserved by small perturbations on real and complex Markov compacts. Applications are given to smooth transformations of polynomial meshes and to polynomial interpolation.2000 AMS subject classification: 30E10, 41A10, 41A63, 65D05.

Polynomial fitting and interpolation on circular sections

“…Among their properties, we quote that WAMs are preserved by affine transformations, can be constructed incrementally by finite union and product, and are ''stable'' under small perturbations [28]. Moreover, we recall that unisolvent interpolation point sets, with slowly (at most polynomially) increasing Lebesgue constant, are WAMs, with CðA n Þ equal to the Lebesgue constant and cardðA n Þ ¼ dimðP d n ðKÞÞ.…”

Trivariate polynomial approximation on Lissajous curves

“…Observe that necessarily card(A n ) ≥ dim(P d n ). Among their properties, we quote that WAMs are preserved by affine transformations, can be constructed incrementally by finite union and product, and are "stable" under small perturbations [29]. It has been shown in the seminal paper [11] that WAMs are nearly optimal for polynomial leastsquares approximation in the uniform norm.…”

“…i.e., the 3-dimensional hyperinterpolation coefficients (18) can be computed by the {c m } and (29). The coefficients of Chebyshev-Lobatto interpolation (28) are at the core of the Chebfun package, cf.…”

Suboptimal Polynomial Meshes on Planar Lipschitz Domains

“…cf., e.g., [6,11,13,15]. Moreover, the polynomial inequality (1) can be relaxed, asking that it holds with C = C n , a sequence of constants increasing at most polynomially with n: in such a case, we speak of weakly admissible polynomial meshes.…”