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.

The half-plane Fourier projection is $(I-i{H}_v)/2$, where $H_v$ is the directional Hilbert transform. A disk of radius $R$, modulated to put a boundary point at the origin, has symbol $1_{\{|\xi +Rv|\le R\}}$, which converges almost everywhere to this half-plane symbol. The Plancherel theorem makes this an $L^2$ multiplier limit.

The inverse Fourier transform of $(1-|\xi |{)}_{+}$ in three dimensions, with $a=2\pi |x|$, is $-4\pi \sin a/a^3+8\pi (1-\cos a)/a^4$. Its leading oscillatory $|x{|}^{-3}$ tail is not absolutely integrable against the radial volume element $4\pi |x{|}^{2}d|x|$. Multiplying by a Gaussian Fourier transform retains a nonzero boundary-derivative tail, so the radial cutoff need not map even Schwartz functions into $L^1$. A tensor product of one-dimensional triangular cutoffs is a valid approximate identity, and must not be confused with this Euclidean radial cutoff.

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.