De Branges' approximation theorem and its generalizations

In 1959 de Branges suggested a solution of Bernstein's problem on weighted polynomial approximation on the real line in terms of the existence of a certain entire function with some special properties. This approach admits applications to other approximation problems (approximation by entire functions of exponential type or by functions from de Branges spaces which appear in spectral theory of differential operators) and turns out especially effective in the problems of stability of the completeness property under small perturbations of the weights. In the talk we will discuss some recent results due to A. Borichev, M. Sodin, H. Woracek and the speaker.