Interlineation of functions of two variables on \(M(M\geq{} 2)\) lines preserving the class \(C^ r(R^ 2)\) (Q1814599)

Let \(L_ q(q=1,\ldots,M)\) be a given family of straight lines of an arbitrary location on the plane \(R^ 2\). The set of all points of intersections in pairs of different lines from \(\{L_ q\}\) is denoted by \(G\), and the normal to \(L_ q\) is denoted by \(n_ q\). It is supposed that on each line \(L_ q(q=1,\ldots,M)\) a family of functions \(f_{qp}\in C^{r-p}(L_ q)\) \((p=0,\ldots,n)\), where \(r>n\), is given. The problem of construction of a function \(f\in C^ r(R^ 2)\) such, that \((\partial/\partial n_ q)^ pf\mid _{L_ q}=f_{qp}\) \((p=0,\ldots,n)\) for all \(q=1,\ldots,M\), is considered. An algorithm for constructing the solution \(f=Q_{Mn}(\{f_{qp}\})\) of the class \(C^ r(R^ 2\backslash G)\) of the above problem is suggested. An integral analogy of the generalized D'Alambert formula is obtained and an integral representation of the remainder of approximation of functions \(u\in C^ r(R^ 2)\) by the operators \(Q_{Mn}(\{(\partial/\partial n_ q)^ pu\mid_{L_ q}\})\) is given.