Recovering fourier coefficients of some functions and factorization of integer numbers

It is shown that if a function defined on the segment $[-1,1]$ has sufficiently good approximation by partial sums of the Legendre polynomial expansion, then, given the function's Fourier coefficients $c_n$ for some subset of $n\in[n_1,n_2]$, one can approximately recover them for all $n\in[n_1,n_2]$. As an application, a new approach to factoring of integers is given.