I purchased 1200 Pokecoins which showed up in the game. Later I discovered they were down to 880, without me ever authorizing any purchases. What is going on?
If you go into the Help Center in the app (Main PokéBall icon : Settings : Help Center), click the "Submit a Request" button in the upper right corner and one of the choices there is "Help with PokéCoins and premium items". That will let you send an e-mail to Niantic about the problem. Report the same thing that you said here. (It's also highly advisable to read the various general and specific FAQ/help entries you'll pass along the way, in case any of them are applicable.)
In theory they may be able to check the logs of shop transactions for your account and figure out what happened. In practice, given that they've really got their hands full with this game, they might just replace the missing coins to make the problem go away.