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

Authors:
Paul C. Kainen;Věra Kůrková;Andrew Vogt
Affiliations:
Department of Mathematics, Georgetown University, Washington, DC 20057-1233, USA;Institute of Computer Science, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 2, 182 07 Prague 8, Czech Republic;Department of Mathematics, Georgetown University, Washington, DC 20057-1233, USA
Venue:
Journal of Approximation Theory
Year:
2007

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Abstract

Quantitative bounds on rates of approximation by linear combinations of Heaviside plane waves are obtained for sufficiently differentiable functions f which vanish rapidly enough at infinity: for d odd and f@?C^d(R^d), with lower-order partials vanishing at infinity and dth-order partials vanishing as @?x@?^-^(^d^+^1^+^@?^), @?0, on any domain @W@?R^d with unit Lebesgue measure, the L"2(@W)-error in approximating f by a linear combination of n Heaviside plane waves is bounded above by k"d@?f@?"d","1","~n^-^1^/^2, where k"d~@pd^1^/^2(e/2@p)^d^/^2 and @?f@?"d","1","~ is the Sobolev seminorm determined by the largest of the L^1-norms of the dth-order partials of f on R^d. In particular, for d odd and f(x)=exp(-@?x@?^2), the L"2(@W)-approximation error is at most (2@pd)^3^/^4n^-^1^/^2 and the sup-norm approximation error on R^d is at most 682(n-1)^-^1^/^2(2@pd)^3^/^4d+1,n=2.