Error bounds for least squares approximation by polynomials

AbstractLet f ϵ Cn+1[−1, 1] and let H[f](x) be the nth degree weighted least squares polynomial approximation to f with respect to the orthonormal polynomials qk associated with a distribution dα on [−1, 1]. It is shown that if ∥qn+1∥∥qn∥ ⩾ max(qn+1(1)qn(1), −qn+1(−1)qn(−1)), then ∥f − H[f]∥ ⩽ ∥fn + 1∥ · ∥qn+1∥∥qn + 1(n + 1)∥, where ∥ · ∥ denotes the supremum norm. Furthermore, it is shown that in the case of Jacobi polynomials with distribution (1 − t)α (1 + t)β dt, α, β > −1, the condition on ∥qn+1∥∥qn∥ is satisfied when either max(α,β) ⩾ −12 or −1 < α = β < −12