 Akhiezer obtained important results in approximation theory (in particular, on extremal problems, constructive function theory, and the problem of moments), where he masterly applied the methods of the geometric theory of functions of a complex variable (especially, conformal mappings and the theory of Riemann surfaces) and of functional analysis.He found the fundamental connection between the inverse problem for important classes of differential and finite difference operators of the second order with a finite number of gaps in the spectrum, and the Jacobi inversion problem for Abelian integrals.This connection led to explicit solutions of the inverse problem for the so-called "finite-gap operators".