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We parametrize the space $\mathcal{Z}$ of Zygmund vector fields on the unitcircle in terms of infinitesimal shear functions on the Farey tesselation. Thenwe express the Hilbert transform and the Fourier coefficients of the Zygmundvector fields in terms of the above parametrization by infinitesimal shearfunctions. Finally, we compute the Weil-Petersson metric on the Teichm\"ullerspace of a punctured surface in terms of shears.

Zygmund vector fields, Hilbert transform and Fourier coefficients in shear coordinates