Question: Can we capture 4K with avconv?

Answer: Yes, you can. Be sure to use the latest avconv build by visiting https://libav.org/ in order to avoid any issues. You can capture a 4K stream with the following command :

>> avconv -f video4linux2 -framerate 30 -video_size 3840×2160 -i /dev/video0 test4k.avi