Thursday, 7 June 2007

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|pp will mainly be those of c1k.



If one does have arbitrary precision, take c1=|c|infty=lim|c|p. To compute this, increase p until you have "enough" significant digits. Then to get c2, subtract c1p from all the |c|pp's and repeat to get c2, and so on.



Obviously, your estimate of ck will be worse than that of ck1, and in double precision I wasn't able to get more than 2 or 3 terms (depends on how large/small the ratios ck1/ck are).

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