Transformers work with AC power, but can you use pulsed DC power to run one?
Yes, you can put pulsed CD through a transformer; that's how a DC-to-DC converter works. (DC to oscillator to transformer to rectifier/regulator.) The efficiency will depend in part on the waveform of the pulses; high frequencies don't go through transformers (or coils generally) very well.
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High frequencies "go through" the right kind of transformer very well
Orkasecurity is correct. But I would SLIGHTLY disagree about high frequencies being used. If possible you would want to use something higher than 60 hertz. Because a higher frequency would give you better rectification and you would need SMALLER electrolytic capacitors due to the higher frequency. However, there is a LIMIT to how high a frequency you can go due to what orkasecurity says, "high frequencies don't go through transformers or coils very well." It is a trade off due to capacitive reactance goes down but inductive reactance goes up as the frequency increases. Certainly you should be able to go to a couple hundred hertz and improve the filtering action of the capacitors without affecting the bad effects of inductive reactance going up. The only reason 60 hertz is commonly used is because that is what comes out of the wall outlet, so you are stuck with it. But if you are making a SWITCHING power supply, you might want to use a higher frequency but not TOO high.