A Fourier spectral method represents a periodic unknown by finitely many Fourier modes and converts differentiation into multiplication by wavenumber.
A spectral truncation retains a finite set of basis modes and sets all omitted coefficients to zero.
For the two-periodic problem
(1)
the Fourier coefficients satisfy the tridiagonal system
(2)
The Fourier–Galerkin method seeks a finite Fourier series whose residual is orthogonal to every retained Fourier mode.

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