I have a panasonic 46" Rear projection TV, model PT-47WX49E from 2001. When I turn on power, I hear it click and then click off after a few seconds. No picture appears on screen. Fuse appears ok.

Hello,
A common problem for these is bad capacitors in the power supply, Remove the back from your Tv and look at the power supply on the left hand side.

You will see roughly around 9 x capacitors that are bulged / domed shaped on top usually these are valued at 3300uf 10V any that have bulged or leaked a brown like substance should be changed.
This will cure the problem as they overheat being sat next to a heatsink that gets very hot!
Try and change them for the same 3300uf but a higher voltage.
Hope this helps