Why is an input capacitor necessary when a battery is used as an input source to an LDO?
Any low-dropout (LDO) regulator requires low source impedance to assure stability. A battery looks fairly reactive at high frequencies; thus a capacitor at the input of the LDO is necessary. Can you use the same 1uF capacitor for three regulators? Yes, as long as the capacitor is physically located within about one cm of the input pin of all of the LDO regulators. This is so that trace impedance does not isolate the capacitor from the input. If that happens, the LDO will not have the low source impedance it needs to be stable. In general, both input and output caps on an LDO should be within one cm of the pin they are connected to.