Local approximation by splines with displacement of nodes

We consider the problem of approximating a function defined on a uniform mesh by the method of local polynomial spline-approximation where the mesh of the nodes of the spline is chosen displaced relative to the mesh of the initial data. Conditions are established for the local form preservation by the spline of the initial data. We study the approximative properties of the method for the case of the simplest local approximation formula and find the optimal values of the displacement parameters.