# Positive sequence

From Encyclopedia of Mathematics

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