Interlineation of functions of 2 variables on M (M≥2) lines with the highest algebraic precision

Abstract

A general algorithm is proposed for constructing interlineation operators$\bar O_{MN} f(x)$, x=(x₁, x₂) with the properties

$$\begin{array}{*{20}c} {\left. {\frac{{\partial ^s \bar O_{MN} f}}{{\partial v_k^s }}} \right|_{\Gamma _k } = \left. {\frac{{\partial ^s f}}{{\partial v_k^s }}} \right|_{\Gamma _k } = \left. {\varphi _{ks} (x)} \right|_{\Gamma _k } ,k = \overline {1,M;} s = \overline {0,N} ,} \\ {\bar O_{MN} x^\alpha \equiv x^\alpha ,0 \leqslant |\alpha | = \alpha _1 + \alpha _2 \leqslant M(N + 1) - 1, x^\alpha = x_1^{\alpha _1 } x_2^{\alpha _2 } ,} \\ \end{array}$$

where {γ_k} is a given set of lines of arbitrary disposition on the plane Ox₁x₂,Ν _k ⊥ γ_k. An integral representation is derived of the residual of approximation of the function f(x) by the operators$\bar O_{MN} f(x)$. Examples are considered of interlineation operators preserving the class C^r(R²), and also operators not preserving the differentiability class, to which the function f(x) belongs.