Conjugate function

Summary

Let $f$ be an integrable function on the circle, and extend it to a harmonic function in the unit disc. The conjugate function $\tilde f$ is the boundary value of the harmonic conjugate of this extension. On Fourier series it acts by $e^{ik\theta} \mapsto -i \sgn(k) e^{ik\theta}$, so it turns $\cos k\theta$ into $\sin k\theta$. It is the circle version of the Hilbert transform. Its $L^p$ bounds control the convergence of Fourier series in $L^p$.