On optimal error bounds for derivatives of interpolating splines on a uniform partition (Q1293273)

In this paper the authors have analyzed the Peano kernel \(N^k_n (u,\nu_n, \theta)\) related to the cardinal spline interpolation problem. Based on the Peano kernel technique, explicit bounds are proved for derivatives of cardinal spline interpolation. The authors have obtained a new representation for this kernel and investigated the nature of zeros. They have applied these results to get exact explicit expressions for \(A^k_n (u,\nu_n)\) and \(A_n^k (\nu_n)\) and bounds for \(A^k_n (u,\nu_n)\) and \(A^k_n (\nu_n)\). They have obtained good bounds for these expressions but exact explicit expressions remain to be found in that case. In short their results are based on new representation of Peano kernel and thorough investigation of their zero distributions. The bounds are given in terms of Euler-Frobenius polynomials and their zeros.