Reconstruction of functions from the traces of their normal derivatives on a line in \(R^ 2\) preserving the class \(C^ r(R^ 2)\) (Q752366)

Let \(f(x,y)\in C^ r({\mathbb{R}}^ 2)\) \(r\geq 0\). We obtain a new integral representation for the error of approximation of the function f with the help of given operators \(L_ nf\) and an estimate of it. This integral representation depends explicitly on the differential operator which is annihilated on functions of the form \(L_ nf(x,y)\), which lets one draw the conclusion of the existence and uniqueness of a solution of the Cauchy problem for the corresponding \((n+1)\)-st order partial differential equation. The solution obtained contains as a special case Taylor's and d'Alembert's formula giving a solution of the Cauchy problem for the equation of a vibrating string.