Optimal algorithms for doubly weighted approximation of univariate functions

We consider a ρ{variant}-weighted L_q approximation in the space of univariate functions f:R+→R with finite ‖f(r)ψ‖Lp. Let α=r-1/p+1/q and ω=ρ{variant}/ψ. Assuming that ψ and ω are non-increasing and the quasi-norm ‖ω‖L1/α is finite, we construct algorithms using function/derivatives evaluations at n points with the worst case errors proportional to ‖ω‖L1/αn-r+(1/p-1/q)+. In addition we show that this bound is sharp; in particular, if ‖ω‖L1/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{variant}=ψ≡1. We also provide a numerical illustration.