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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!

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    $\begingroup$ You're going to struggle to recover individual terms of the sequence, since re-ordering the sequence doesn't change any of the l_p norms. You might have better luck approximating the function C(x) = (number of n such that |c_n| > x); you're essentially trying to recover C(x) from its Mellin transform. $\endgroup$
    – David Loeffler
    Commented Mar 9, 2010 at 18:07
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    $\begingroup$ of course you're right - I should require that c_n be monotone. I'll look into your suggestion, thank you. $\endgroup$
    – MarkV
    Commented Mar 9, 2010 at 18:13

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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).

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    $\begingroup$ Translate your problem to trying to determine a probability distribution from its moments. Then Google "moment problem". $\endgroup$
    – Bill Johnson
    Commented Mar 10, 2010 at 17:19

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