We prove the following Whitney estimate. Given 0 < p ≤ ∞, r ∈ ℕ, and d ≥ 1, there exists a constant C(d, r, p), depending only on the three parameters, such that for every bounded convex domain Ω ⊂ ℝd, and each function f ∈ Lp(Ω), E r-1 (f, Ω)p ≤ C(d, r, p)ωr (f, diam(Ω))p, where Er-1 (f, Ω)p is the degree of approximation by polynomials of total degree, r - 1, and ωr(f, ·)p is the modulus of smoothness of order r. Estimates like this can be found in the literature but with constants that depend in an essential way on the geometry of the domain, in particular, the domain is assumed to be a Lipschitz domain and the above constant C depends on the minimal head-angle of the cones associated with the boundary. The estimates we obtain allow us to extend to the multivariate case, the results on bivariate Skinny B-spaces of Karaivanov and Petrushev on characterizing nonlinear approximation from nested triangulations. In a sense, our results were anticipated by Karaivanov and Petrushev.
- John's theorem
- Nonlinear approximation
- Piecewise polynomial approximation
- Whitney estimates