Stability estimate in the problem of reconstruction of an analytical characteristic function (Q1175847)

Let \(F(x)\) be a given distribution function in \(\mathbb{R}^1\) having finite moments of all orders, which can be uniquely reconstructed by these moments. Assume that \(f(t)\) is the characteristic function of \(F(x)\), and \(G(x)\) is some distribution function with the characteristic function \(g(t)\). If \(f(t)\) and \(g(t)\) coincide in some neighborhood of the point \(t=0\), then the functions \(F(x)\) and \(G(x)\) coincide. This paper is devoted to the solution of the following problem. Let \(f(t)\) and \(g(t)\) coincide only approximately, i.e. \(| f(t)-g(t)|\leq\varepsilon\) for all \(t\in [-T_0,T_0]\) (here \(T_0\) is a fixed number). What can we say about the proximity of the distributions \(F\) and \(G\)?