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

• Autores: Paul C. Kainen, Vera Kurková, Andrew Vogt
• Localización: Journal of approximation theory, ISSN 0021-9045, Vol. 147, Nº 1, 2007, págs. 1-10
• Idioma: inglés
• DOI: 10.1016/j.jat.2006.12.009
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• Resumen

  • 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+), >0, on any domain with unit Lebesgue measure, the L2(O)-error in approximating f by a linear combination of n Heaviside plane waves is bounded above by kdfd,1,8n-1/2, where kd(pd)1/2(e/2p)d/2 and fd,1,8 is the Sobolev seminorm determined by the largest of the L1-norms of the dth-order partials of f on . In particular, for d odd and f(x)=exp(-x2), the L2(O)-approximation error is at most (2pd)3/4n-1/2 and the sup-norm approximation error on is at most .