Jackson-type theorems on some transcendental curves in [formula omitted

Abstract

“Efficient curves” in the sense of the best rate of multivariate 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 , e t ) : 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 R d and show that they too are efficient. Transcendental curves of the form { ( t , t λ ) : δ ⩽ t ⩽ 1 } are shown to be efficient for δ > 0 , contradicting another claim of Newman and Raymon.