The energy expense of superoscillations

For any fixed bandwidth, there are finite energy signals which oscillate arbitrarily fast on arbitrarily long time intervals. More precisely, for any fixed bandwidth, it is possible to find finite energy signals f which at arbitrarily chosen times {t
_i
}
_i=1
^N
possess arbitrarily prescribed amplitudes a
_i
, i.e. which obey f(t
_i
) = a
_i
for i = 1,2, ..., N where N is arbitrarily large. This paper investigates to which extent such superoscillating signals could be used for the fast transmission of information through low bandwidth channels. The main result is that for fixed noise level, arbitrary amounts of information can be compressed into arbitrarily short segments of signals of arbitrarily low bandwidth, without having to increase the level of the amplitudes which encode the information. The price to be paid is that, for fixed message size, the energy expense grows polynomially with the compression and that, for fixed compression, the energy expense grows exponentially with the message size.