Forward and Converse Theorems of Polynomial Approximation for Exponential Weights on [−1, 1], I Original Research Article

We consider exponential weights of the formw≔e−Qon (−1, 1) whereQ(x) is even and grows faster than (1−x2)−δnear ±1, someδ>0. For example, we can takeQ(x)≔expk((1−x2)−α), k⩾0, α>0,where expkdenotes thekth iterated exponential and exp0(x)=x. We prove Jackson theorems in weightedLpspaces with norm ‖fw‖Lp(−1, 1)for all 0