On Multivariate Incomplete Polynomials on Starlike Domains

Kroó, András and Szabados, József (2014) On Multivariate Incomplete Polynomials on Starlike Domains. CONSTRUCTIVE APPROXIMATION, 39 (2). pp. 397-419. ISSN 0176-4276

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Abstract

Multivariate incomplete polynomials are considered on compact 0-symmetric starlike domains. Problems of density and quantitative approximation properties of such polynomials are investigated. It is shown that density holds for a certain class of starlike domains which includes both convex and some nonconvex domains. On the other hand, a family of nonconvex starlike domains is also found for which density fails. In addition, it is also shown that on 0-symmetric convex bodies in {Mathematical expression}, continuous functions can be approximated by θ-incomplete polynomials with the rate O(ω2(n-1/(d+3))). Moreover, if the convex body is the intersection of simplexes with vertex at the origin, then this order improves to {Mathematical expression}. © 2013 Springer Science+Business Media New York.

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