On an optimal interpolation formula with derivatives in the Sobolev space

Abstract

In this work, the problem of constructing an optimal interpolation formula involving derivatives is studied. The values of the unknown function are required not only at the nodal points but also the values of its first three derivatives at these nodes. An upper bound for the interpolation error is obtained using an extremal function, whose explicit form is determined. Furthermore, the squared norm of the corresponding error functional is derived. Since this norm depends on the coefficients, the Lagrange function is introduced, and its partial derivatives with respect to the coefficients are computed and set equal to zero, leading to a system of equations. The resulting system is solved by the method proposed by Sobolev. To this end, the discrete analogue of the differential operator $\frac{d^2}{dx^2}$ is employed to solve the system and to determine the coefficients of the interpolation formula.