Approximations of the exponential function and relative closeness of stable signals

The results presented here are based on the use of approximations of analytic functions in certain questions involving numerical methods for “physical” equations. The concepts of relative closeness and of stability of signals are introduced. A simple method is given for approximate inversion of the Laplace transformation, with an estimate of the relative error. Also, estimates are obtained for the dimensions of spaces in which it is possible to find approximate solutions of multidimensional systems of linear equations, as a function of the accuracy of approximation and the “size” of the spectrum of the operator.