Recovery of integrable functions and trigonometric series

Classes Gamma of L-1-functions with fixed rate of decrease of their Fourier coefficients are considered. For each class Gamma, it is shown that there exists a (recovery) set G with arbitrarily small measure such that any function in Gamma can be recovered from its values on G. A formula for evaluation of the Fourier coefficients of such a function from its values on G is given. In addition, it is shown that, for any L-1-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 G is also solved.