but if its using windows software to capture the screen its probobly not going to work (it might on wine apps but not the whole desktop)......im guessing a CRUDE hack might be to run a VNC server on your linux box and install a windows VNC client via wine and have it logged into yourself (like I said very crude)....your best bet probobly would be to get a VGA/HDMI/DVI (whichever one your using) extender and plug it into the tv.
