Error bounds for least squares approximation by polynomials (Q1262487)

Let \(f\in C^{n+1}[-1,1]\) and \(H[f](x)=\sum^{n}_{k=0}q_ k(x)\int^{1}_{-1}f(t)q_ k(t)d\alpha (t)\) be the weighted least squares approximation to f with respect to orthonormal polynomials \(\{q_ k\}\) associated with the distribution \(d\alpha\) on the interval [-1,1]. If \(q_ n\) and \(q_{n+1}\) are such that \(\bar q_{n+1}/\bar q_ n\geq \max (q_{n+1}(1)/q_ n(1),\quad -q_{n+1}(-1)/q_ n(-1))\) then \(\| f-H[f]\| \leq \| f^{(n+1)}\| \cdot \| q_{n+1}\| /\| q_{n+1}^{(n+1)}\|\) where \(\| \cdot \|\) denotes the supremum norm on [-1,1]. This result is used in the case of Jacobi polynomials with distribution \((1-t)^{\alpha}(1+t)^{\beta}dt,\) \(\alpha,\beta >-1\).