Question

I'm wondering if there is a way to approximate the first $M$ terms of a non-increasing $\ell^2$ sequence ${c_n}$ if we know $|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$?

I've tried truncating the sequence at $c_{M+Q}$ for $Q>1$, and applying Newton's method the the $M+Q$-dimensional system of equations $|c|_p^p = \sum^{M+Q} c_n^p$, $p=2,\dots,M+Q+1$, but the jacobian is ill conditioned and this blows up. Is there a better way? Do you know of any work done on this problem?

Thanks!

Answer 1

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).