Recovering Fourier coefficients of modular forms and factoring of integers

TL;DR

This paper presents a method to recover Fourier coefficients of functions from partial data and applies this technique to develop a novel approach for integer factorization.

Contribution

It introduces a new technique for reconstructing Fourier coefficients of functions from partial information and applies it to integer factorization.

Findings

Successful recovery of Fourier coefficients from partial sums

New approach to integer factorization based on Fourier analysis

Potential for improved algorithms in computational number theory

Abstract

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 may approximately recover them for all $n\in[n_1,n_2]$. As an application, a new approach to factoring of integers is given.