Approximation of Sobolev classes by polynomials and ridge functions

Authors:
V. N. Konovalov;D. Leviatan;V. E. Maiorov
Affiliations:
Institute of Mathematics, NAS of Ukraine, 01601 Kyiv, Ukraine;Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978 Tel Aviv, Israel;Technion I.I.T., 32000 Haifa, Israel
Venue:
Journal of Approximation Theory
Year:
2009

Citing 6
Cited 1

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A Sobolev-type upper bound for rates of approximation by linear combinations of Heaviside plane waves

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Abstract

Let W"p^r(B^d) be the usual Sobolev class of functions on the unit ball B^d in R^d, and W"p^@?^,^r(B^d) be the subclass of all radial functions in W"p^r(B^d). We show that for the classes W"p^@?^,^r(B^d) and W"p^r(B^d), the orders of best approximation by polynomials in L"q(B^d) coincide. We also obtain exact orders of best approximation in L"2(B^d) of the classes W"p^@?^,^r(B^d) by ridge functions and, as an immediate consequence, we obtain the same orders in L"2(B^d) for the usual Sobolev classes W"p^r(B^d).