Periodic Fourier coefficient

For a $\mathbb Z^d$ -periodic function $f$ integrable on $[0,1]^d$ , its Fourier coefficient at $m\in\mathbb Z^d$ is

\widehat f(m)=\int_{[0,1]^d} f(x)\overline{e_m(x)}\,dx =\int_{[0,1]^d}f(x)e^{-2\pi i m\cdot x}\,dx.

The integral uses a unit-volume cell, and $e_m$ is the Fourier character. The coefficient satisfies $|\widehat f(m)|\le\|f\|_{L^1([0,1]^d)}$ .

Other periods and real values

For a scalar $2\pi$ -periodic function, $\widehat f(m)=(2\pi)^{-1}\int_0^{2\pi}f(\theta)e^{-im\theta}\,d\theta$ . If $f$ is real-valued, conjugating the integral gives $\widehat f(-m)=\overline{\widehat f(m)}$ . Vector and matrix coefficients are defined componentwise.

References

Leonid Ryzhik, Lecture notes for Math 205A, Chapters 8–9.