Fourier coefficient

Summary

The $n$-th Fourier coefficient $\hat f(n)$ of a $2\pi$-periodic integrable function $f$ measures how much of the wave $e^{inx}$ is present in $f$. It is the average of $f(x) e^{-inx}$ over one period, and it is the inner product of $f$ with the Fourier monomial $e_n$. The sequence $(\hat f(n))_{n \in \Z}$ determines $f$ up to a null set, and the Fourier series of $f$ is $\sum_n \hat f(n) e^{inx}$. The size of the coefficients for large $\abs{n}$ reflects the smoothness of $f$.