Half-plane Fourier projection
= Half-plane Fourier projection
{title2=$\widehat{P_vg}(\xi)=\mathbf1_{\{\xi\cdot v<0\}}\widehat g(\xi)$}
The half-plane Fourier projection is $(I-iH_v)/2$, where $H_v$ is the <directional Hilbert transform>. A disk of radius $R$, modulated to put a boundary point at the origin, has symbol $\mathbf1_{\{|\xi+Rv|\le R\}}$, which converges almost everywhere to this half-plane symbol. The <Plancherel theorem> makes this an $L^2$ multiplier limit.