Take a look at the datasheet for an LM723 regulator. This is a linear regulator that will require pass transistors that will handle the current requirements. Stuff I am used to seeing uses 2N3055 transistors. This is pretty well established circuitry and there are plenty of other ways to do the job including using switching regulators.
I don't know if the 48 volts RMS from the transformer is enough if you really want 48 vdc and want it with no ripple. With no load the capacitor will charge, as you suggest, to the peak voltage however as you load the supply the voltage will drop. If you use a full wave bridge the voltage available to charge the capacitor never goes to zero but at what point it will drop requires some math. Someone else can probably offer some guidance.