Abstract :
“Efficient curves” in the sense of the best rate ofmultivariate polynomial approximation to contractive
functions on these curves, were first introduced by D.J. Newman and L. Raymon in 1969. They
proved that algebraic curves are efficient, but claimed that the exponential curve γ := {(t , et ): 0
t 1} is not. We prove to the contrary that this exponential curve and its generalization to higher
dimensions are indeed efficient. We also investigate helical curves in Rd and show that they too are
efficient. Transcendental curves of the form {(t , tλ): δ t 1} are shown to be efficient for δ > 0,
contradicting another claim of Newman and Raymon.
2004 Elsevier Inc. All rights reserved
