Polynomial approximation on C^2-domains
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We introduce appropriate computable moduli of smoothness to characterize the rate of best approximation by multivariate polynomials on a connected and compact C^2-domain Ω⊂ℝ^d. This new modulus of smoothness is defined via finite differences along the directions of coordinate axes, and along a number of tangential directions from the boundary. With this modulus, we prove both the direct Jackson inequality and the corresponding inverse for the best polynomial approximation in L_p(Ω). The Jackson inequality is established for the full range of 0<p≤∞, while its proof relies on a recently established Whitney type estimates with constants depending only on certain parameters; and on a highly localized polynomial partitions of unity on a C^2-domain which is of independent interest. The inverse inequality is established for 1≤ p≤∞, and its proof relies on a recently proved Bernstein type inequality associated with the tangential derivatives on the boundary of Ω. Such an inequality also allows us to establish the inverse theorem for Ivanov's average moduli of smoothness on general compact C^2-domains.
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