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

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 , with lower-order partials vanishing at infinity and dth-order partials vanishing as ∥x∥-(d+1+ɛ), ɛ>0, on any domain Ω⊂Rd with unit Lebesgue measure, the -error in approximating f by a linear combination of n Heaviside plane waves is bounded above by kd∥f∥d,1,∞n-1/2, where kd∼(πd)1/2(e/2π)d/2 and ∥f∥d,1,∞ is the Sobolev seminorm determined by the largest of the -norms of the dth-order partials of f on Rd. In particular, for d odd and f(x)=exp(-∥x∥2), the -approximation error is at most (2πd)3/4n-1/2 and the sup-norm approximation error on Rd is at most .