Are there any astronomical examples where the transverse doppler effect(Horizontal doppler effect) is applied (Derives a meaningful result)?
Accounting for the transverse Doppler effect (and other relativistic effects) is essential in modelling the X-ray spectral emission lines from the accretion discs around black holes (e.g. Cadaz & Calvani 2005). In this case the transverse Doppler effect is "mixed up" with gravitational redshift and it is treated holistically in the Schwarzschild or Kerr metrics.
The transverse Doppler effect is also essential in interpreting the signals from binary pulsars and even in single pulsars because of the rotation of the Earth and its motion around the Sun.