Optimal algorithms for doubly weighted approximation of univariate functions

We consider a ϱϱ-weighted LqLq approximation in the space of univariate functions f:R+→Rf:R+→R with finite ‖f(r)ψ‖Lp‖f(r)ψ‖Lp. Let α=r−1/p+1/qα=r−1/p+1/q and ω=ϱ/ψω=ϱ/ψ. Assuming that ψψ and ωω are non-increasing and the quasi-norm ‖ω‖L1/α‖ω‖L1/α is finite, we construct algorithms using function/derivatives evaluations at nn points with the worst case errors proportional to ‖ω‖L1/αn−r+(1/p−1/q)+‖ω‖L1/αn−r+(1/p−1/q)+. In addition we show that this bound is sharp; in particular, if ‖ω‖L1/α=∞‖ω‖L1/α=∞ then the rate n−r+(1/p−1/q)+n−r+(1/p−1/q)+ cannot be achieved. Our results generalize known results for bounded domains such as [0,1][0,1] and ϱ=ψ≡1ϱ=ψ≡1. We also provide a numerical illustration.