I'm looking for detailed explanation (or even proof) of how to
reconstruct a distribution from full set of moments.

I've found this page so far, but it doesn't contain a proof and it's
also unclear about exact conditions:
http://www.plmsc.psu.edu/~www/matsc597/probability/moments/node3.html#SECTION00030000000000000000

This should be fairly classical mathematical result.

Hi borel

The 'missing link' is

SEARCH TERMS: The Characteristic Function of a Probability Distribution

Its Taylor series gives you moments
The Characteristic Function as a Moment-Generating Function

it's Fourier tranfrom is the distribution
http://www2.sjsu.edu/faculty/watkins/charact.htm

see also
http://en.wikipedia.org/wiki/Characteristic_function
http://mathworld.wolfram.com/UniformDistribution.html

cumulant
http://mathworld.wolfram.com/UniformDistribution.html

etc. Hope this explains the  one-to-one correspondence

Hedgie

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Request for Answer Clarification by borel-ga on 02 Apr 2006 23:47 PDT

Thank you for these links, they seem to be pointing to the right directions.

However, I was hoping for more detailed information: existence,
uniqueness, more detailed conditions in general etc. Something that is
more "formal" (in mathematical sense).

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Clarification of Answer by hedgie-ga on 19 Dec 2006 12:17 PST

Thanks for the rating  Borel.

For the issues you mention  existence .., I suggest to combine the
corresponding theorems for the Taylor expansion and Fourier Transform.

Although these links are not exactly what I was hoping for, they have
pointed me to the right direction. The detailed answer I hoped for is
obviously only in textbooks and not on the internet.

Thank you Hedgie.

Well I'm guessing you will have a hard time finding the answer to this
question in a formal proof at least.  My final stats course book (in a
stats major) gives the theorem that if the moment generating functions
for distributions X and Y are equal (mx(t)=my(t)) for all values of t
then X and Y have the same prob. dist. However it adds "the proof of
this theorem is beyond the scope of this text"  Now having all the
moments to a function seems like a hard task in most cases for me, as
many cases there are infinite moments.  However having the m.g.f. in
closed form allows you of course to generate any moment you want. 
Thusly if the mgfs are equal all moments must be equal, from this we
know that E{Y^k)=E(X^k) for all k such that k is in Z+  Where to go
from there is where I get lost, obviously intuition leads you to Y=X
but intuition doesn't amount to much.