I'm not sure the regulator will like to have a capacitor at the output and a full ripple at the input. I've learned (30 years ago) you have to put a diode between the in/out pins of linear reg to protect them for any situation where there is a lower voltage at the input vs the output. Also that may explain the waveform you have.

Anyway, take a look a this very interesting circuit:

http://www.dextrel.net/diyzerocrosser.htm