Fourier multiplier ($m(\xi )$)

A Fourier multiplier is a function $m$ on frequency space used to define an operator by multiplication after taking a Fourier transform.

A Fourier multiplier operator $T_m$ is defined by $\widehat{T_{m}f}(\xi )=m(\xi )\hat{f}(\xi )$. On $L^2$, a bounded symbol gives a bounded operator by the Plancherel theorem, while an unbounded symbol defines a closed linear operator on the functions for which $m\hat{f}$ remains square-integrable.