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Positive sequence

From Encyclopedia of Mathematics
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A sequence of real numbers in the interval such that for any polynomial

that is not identically zero and is not negative on the expression

If for any such polynomial , then the sequence is called strictly positive. For the sequence in to be positive, the existence of an increasing function on for which

(1)

is necessary and sufficient.


Comments

A (strictly) negative sequence can be similarly defined and has a similar property. The problem of deciding whether for a given sequence of real numbers there is a positive Borel measure on such that is known as the Hamburger moment problem. The condition (1) is a moment condition, cf. Moment problem.

References

[a1] H.J. Landau (ed.) , Moments in mathematics , Amer. Math. Soc. (1987) pp. 56ff
How to Cite This Entry:
Positive sequence. M.I. Voitsekhovskii (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Positive_sequence&oldid=12427
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098