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The data sheet usually states the capacitor size range for input and output. Example: 'Cout needs to be larger than 0.1uF'
The inductor is not needed but helps with high frequencies. The ripple rejection is graphed from 100hz to 10khz. At about 5khz and above the rejection gets increasingly worse. The output LC resonant frequency could be set to 10khz to 100khz to help with HF noise. Some companies show a graph of the response of the regulator with different amount of capacitors.
Cin is mostly a function of where you are getting the power from. (50/60hz ripple needs larger capacitors)
Types of caps matter as well, lower ESR the better. Except in certain regulators the output
cap can be speced in datasheet as needing a minimum to insure stability. This is true
of many of the older LDO regulators with PNP infernal pass transistors. So always look for
this topic in the datasheet.
If we were talking about a switching regulator, which we are not for this thread. The two part answer would be:
The inductor would be chosen to handle a particular current ripple for a nominal load and to perhaps guarantee continuous conduction mode under those conditions.
The capacitor would be chosen to achieve a desired voltage ripple under those condition.
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