Recovering fourier coefficients of some functions and factorization of integer numbers

It is shown that if a function determined on the segment −1, 1] has a sufficiently good approximation by partial sums of its expansion over Legendre polynomial, then, given the function’s Fourier coefficients c _n for some subset of n ∈ n ₁, n ₂], one can approximately recover them for all n ∈ n ₁, n ₂]. A new approach to factorization of integer numbers is given as an application.