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

For an integrable function of period $L>0$, its complex Fourier coefficients measure its inner product with each complex exponential mode:For a real function of period $2\pi$, the real coefficient convention of Fourier series gives ${\hat{f}}_0=a_0/2$ and ${\hat{f}}_n=(a_n-i{b}_n)/2$ for $n\ge 1$, with ${\hat{f}}_{-n}=\overline{{\hat{f}}_n}$. For $f(x)=1/2+\cos x+\frac{1}{2}\sin (2x)$, the only nonzero complex Fourier coefficients are ${\hat{f}}_0=1/2$, ${\hat{f}}_{\pm 1}=1/2$, ${\hat{f}}_2=-i/4$, and ${\hat{f}}_{-2}=i/4$. This explains why the real and complex lists look different while representing the same function.

The rate at which Fourier coefficients approach zero as their index increases. The differentiability of the periodic extension, including matching values and derivatives at its joins, controls this rate through integration by parts. An absolutely bounded tail $O(n^{-p})$ with $p>1$ gives a uniform truncation error $O(N^{1-p})$.

For an integrable function on $[-L,L]$, with $L>0$ and integer $n\ge 1$, the displayed Fourier coefficient multiplies the mode $\sin (n\pi x/L)$ in its real Fourier series. Fourier orthogonality fixes the normalization because the squared mode has integral $L$. If $f$ is odd, the product with the sine is even, so $b_n=(2/L){\int }_0^{L}f(x)\sin (n\pi x/L)dx$. These are also the coefficients of the Fourier sine series of the restriction to $(0,L)$.

If $f$ is integrable and $2\pi$-periodic and $n$ is a positive integer, then ${\int }_0^{2\pi }f(nx)e^{imx}dx=0$ for every integer $m$ with $0<|m|<n$. Split the integral into $n$ equal intervals and substitute $y=nx-2\pi j$ to obtainThe root of unity sum vanishes by the finite geometric series formula. The zero mode is $\int f$, so a mean-zero dilated function is orthogonal to all trigonometric polynomials of degree at most $n-1$.

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 $L$,