Optimal algorithms for doubly weighted approximation of univariate functions.

We consider a ϱ -weighted L q approximation in the space of univariate functions f : R + → R with finite ‖ f ( r ) ψ ‖ L p . Let α = r − 1 / p + 1 / q and ω = ϱ / ψ . Assuming that ψ and ω are non-increasing and the quasi-norm ‖ ω ‖ L 1 / α is finite, we construct algorithms using function/derivatives evaluations at n points with the worst case errors proportional to ‖ ω ‖ L 1 / α n − r + ( 1 / p − 1 / q ) + . In addition we show that this bound is sharp; in particular, if ‖ ω ‖ L 1 / α = ∞ then the rate n − r + ( 1 / p − 1 / q ) + cannot be achieved. Our results generalize known results for bounded domains such as [ 0 , 1 ] and ϱ = ψ ≡ 1 . We also provide a numerical illustration. [ABSTRACT FROM AUTHOR]