Recovery of integrable functions and trigonometric series

Abstract

Abstract Classes of -functions with fixed rate of decrease of their Fourier coefficients are considered. For each class , it is shown that there exists a (recovery) set with arbitrarily small measure such that any function in can be recovered from its values on . A formula for evaluation of the Fourier coefficients of such a function from its values on is given. In addition, it is shown that, for any -function, a function-specific recovery set can be found. The problem of recovery of general trigonometric series from the Zygmund classes which converge to summable functions on such sets is also solved. Bibliography: 10 titles.