Fourier coefficient (${\hat{f}}_n$)

For a function of period $L$, its complex Fourier coefficients are

For a periodic absolutely continuous function with derivative in $L^2$, integration by parts gives $\widehat{f^{\prime }}(n)=in\hat{f}(n)$. Bessel's inequality and the Cauchy-Schwarz inequality prove the displayed bound in the normalized circle norm. Adding the constant coefficient gives an absolutely and uniformly convergent Fourier series. Merely bounding individual coefficients by $O(1/|n|)$ would not establish absolute convergence.

If a periodic integrable function has all Fourier coefficients zero, it vanishes almost everywhere. Its Fejér sums vanish, whereas the Fejér kernel is an approximate identity and converges to that function in the integrable norm. Applied to a periodized squared Fourier transform, this detects orthogonality of integer translates.

If a periodic function extends analytically and remains bounded in a nonzero complex strip around the real axis, contour shifting gives constants $C,c>0$ such thatThis exponential decay underlies spectral convergence of Fourier approximation and periodic trapezoidal quadrature.

When regularity and convergence justify termwise differentiation, the derivative of a Fourier mode multiplies its coefficient by its frequency. For period $L$,