Are there experiments that measure relativistic Doppler effect? Relativistic Doppler effect should be different from regular Doppler effect since there is also time dilation.
Are there any experiments that prove that ?
 A: The relativistic Doppler effect has “classical” and due to dilation of moving clock – “relativistic” components.
Transverse Doppler effect is purely relativistic effect and is in accordance with dilation of moving clock.
The Ives - Stilwell experiment tested the contribution of relativistic time dilation and was the first quantitative confirmation of time dilation.
https://en.wikipedia.org/wiki/Ives%E2%80%93Stilwell_experiment
Transverse Doppler Effect has also been confirmed by Mossbauer rotor experiment in centrifuge. In these experiments either absorber rotates around the source of radiation or vice versa.
Among notable experiments there are the following:
Kuendig, Hay at All, Champeney and Moon.
Hay H J, Schiffer J P, Cranshaw T E and Egelstaff P A 1960 Phys. Rev. Lett. 4 165-6
Kuendig, W. (1963). MEASUREMENT OF THE TRANSVERSE DOPPLER EFFECT IN AN ACCELERATED SYSTEM. Physical Review (U.S.) Superseded in Part by Phys. Rev.
In Champeney and Moon (1963) experiment a source and an absorber were placed on the opposite sides of the rim.
http://iopscience.iop.org/article/10.1088/0370-1328/77/2/318/meta
Quite recently – Kholmetskii, Missevitch, Yarman
https://arxiv.org/ftp/arxiv/papers/1503/1503.05853.pdf
Related polemical articles:
Essen J 1964 Nature 202 787
Jennison R C 1964 Nature 203 395-6
E. Zanchini "Correct Interpretation of two experiments on the Transverse Doppler Shift". Phys. Scr. 86 (2012)
Essen L Bearing on the recent experiments on the special and general theories of relativity No 4934 Nature, May 23 1964
Essen L A time dilatation experiment based on the Mossbauer effect. Proc. Phys. Soc.vol.86 1965
Quite good and complete explanation of Relativistic Doppler Effect
http://www.mathpages.com/home/kmath587/kmath587.htm
