Finite Hilbert transform

Summary

The finite Hilbert transform sends a function $f$ on the interval $(-1, 1)$ to the Cauchy principal value integral $\frac{1}{\pi}\, \mathrm{PV}\!\int_{-1}^{1} \frac{f(t)}{t - x}\, dt$. Up to sign, it is the Hilbert transform of $f$ extended by zero, looked at only on the interval. It is the singular part of most singular integral equations on an arc, such as the airfoil equation. Unlike the Hilbert transform on the line, it is not invertible on $L^p(-1, 1)$ for any $1 < p < \infty$. For $1 < p < 2$ it even sends a nonzero function to $0$.