Estimates of solutions to infinite systems of linear equations and the problem of interpolation by cubic splines on the real line

We study the solvability of bi-infinite systems of linear equations whose matrices are diagonally dominant. We prove that the estimates of the norm of the solution in terms of the diagonal dominance value well-known in the case of finite systems of linear equations are also valid for bi-infinite systems of equations. The estimates are used in interpolation by splines on nonuniform meshes on the real line. Using the estimates, we prove the existence and uniqueness of a cubic spline of linear or quadratic growth interpolating data of linear or quadratic growth, without any constraints on node spacing. The familiar estimates of the interpolation error on a segment are carried over to the case of interpolation on the whole real line.