Periodic Fourier multiplier

A periodic Fourier multiplier with symbol $a:\mathbb Z^d\to\mathbb C$ acts on a finite Fourier sum by

T_a\left(\sum_m c_m e_m\right)=\sum_m a(m)c_m e_m.

It is a linear operator diagonal in the Fourier characters. Extending it to a function space requires a convergence and boundedness statement.

Standard symbols

The derivative $\partial_{x_j}$ has symbol $2\pi i m_j$ ; the Laplacian has symbol $-4\pi^2|m|^2$ . Subtracting the mean has symbol zero at $m=0$ and one elsewhere. For a constant direction $v$ , formally inverting $v\cdot\nabla$ uses $(2\pi i v\cdot m)^{-1}$ at nonzero denominators. Resonances and small denominators require additional analysis.