Abstract: | We prove the following Whitney estimate. Given 0 < p \le \infty,
r \in N, and d \ge 1, there exists a constant C(d,r,p),
depending only on the three parameters, such that for every
bounded convex domain \subset Rd, and each function
f \in Lp( ),
Er-1(f, )p \le C(d,r,p) r(f, diam( ))p,
where Er-1(f, )p is the degree of
approximation by polynomials of total degree, r – 1, and
r(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. |