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How can a star's rotation speed be measured?
If stars could be resolved, then you could use the Doppler shift to see that half of the star is bluer and the other half is redder, but stars (other than our Sun) are seen as point objects. With only one point of measurement, how can the rotation be inferred?
This article mentions that stellar rotation speeds have been measured, but doesn't explain how.
The projected equatorial velocity of a star (commonly written $v \sin i$, where $i$ is an inclination angle for the rotation axis to the line of sight) is measured by observing the Doppler broadening effect on spectral lines.
There is no need to resolve the star. Light from one limb is approaching at $+v\sin i$, whilst light from the other limb is receding at $-v\sin i$. The net effect is that the spectral lines are convolved with a broadening function that has a full width of $2 v\sin i$.
Providing that this rotational broadening makes a significant contribution to the overall line width, then it can be measured (by direct fitting, Fourier transform or cross-correlation methods).
True rotation speed requires a measurement of rotation period and knowledge of the stellar radius. Whilst the former is known, the latter usually isn't.
Note in your article, I suspect the rotation speeds are actually the inverse of rotation periods. Angular velocities are often referred to as rotation rates and can be derived without spectroscopy by measuring periodic light variations from surface inhomogeneities on a star.
