The Lagrange interpolation polynomials in Sobolev spaces (Q1390466)

Using the notations of the paper, being the common ones, the authors proved, among others, the following theorems. Theorem 1. For an arbitrary function \(x(s)\in W^r_2\), the interpolation polynomials \({\mathcal L}_n(x; s)\) converge in the space \(W^r_2\) for \(r> 1/2\), and for \(j= 1,2,3\) \[ E_n(x)_{(j)}\leq\| x-{\mathcal L}_nx\|_{(j)}\leq A_j E_n(x)_{(j)}, \] \[ E_n(D^rx)_2\leq\| x-{\mathcal L}_n x\|_{(j)}\leq B_j E_n(D^rx)_2, \] hold, where \(A_j\), \(B_j\) are given exactly, furthermore the norms \(\|\cdot\|_j\) are the most usable ones, being equivalent if \(r> 1/2\). Theorem 3. For any \(x(s)\in W^\ell_2[0, 2\pi]\), \(\ell\geq r> 1/2\), \({\mathcal L}_n(x; x)\) converge in \(W^r_2[0, 2\pi]\) with the rate \[ \| x-{\mathcal L}_n x\|_{W^r_2}= O(n^{r-\ell} E^T_n(x^{(\ell)})_2). \] Similar results are also proved for algebraical interpolation polynomials.