Interlineation of Functions

This chapter is devoted to the construction and study of interlineation operators of functions of two or more variables with and without preservation of the class of differentiation. When constructing interlineation operators of functions, new information operators are used: traces of an approximate function and traces of some differential operators with partial derivatives up to the specified order on a system of parallel or intersecting lines. If information operators use only traces of functions, then we have an interlineation of the Lagrangian type. If the interlineation of functions uses traces of the approximate function and traces of some system of differential operators from the approximate function, then we have a Hermitian-type interlineation. If the approximate function has continuous derivatives only up to a certain order, then Hermitian interlineation operators with given traces on the system of lines give functions that do not have the same class of differentiation as the approximate function. For this case, the work presents a generalized D'Alembert formula (the classical D'Alembert formula is its partial case), which automatically preserves the same class of differentiation to which the approximate function belongs. The work presents examples of Hermitian interlineation operators with and without preservation of the class of differentiation on the system of parallel, mutually perpendicular, and intersecting lines. Here give integral representations of the residual terms of the approximation of differentiation functions of two variables with and without preservation of the class of differentiation. An example of Lagrangian interlineation on a system of mutually perpendicular lines with the optimal choice of these interlineation lines is given.