na.numerical analysis - Can we find an l-2 sequence if we know all l-p norms?

Sorry to answer my own question, but having thought about it more, I realize this is impossible unless you have arbitrarily high precision for all the $|c|_p$'s. The reason is, as $p$ grows, the leading digits of $|c|_p^p$ will mainly be those of $c_1^k$.

If one does have arbitrary precision, take $c_1 = |c|_infty =lim |c|_p$. To compute this, increase $p$ until you have "enough" significant digits. Then to get $c_2$, subtract $c_1^p$ from all the $|c|_p^p$'s and repeat to get $c_2$, and so on.

Obviously, your estimate of $c_k$ will be worse than that of $c_{k-1}$, and in double precision I wasn't able to get more than 2 or 3 terms (depends on how large/small the ratios $c_{k-1}/c_k$ are).