Positive sequence

A sequence $ \mu _ {0} , \mu _ {1} \dots $ of real numbers in the interval $ [ a, b] $ such that for any polynomial

$$ P( x) = a _ {0} + a _ {1} x + \dots + a _ {n} x ^ {n} $$

that is not identically zero and is not negative on $ [ a, b] $ the expression

$$ \Phi ( P) = a _ {0} \mu _ {0} + a _ {1} \mu _ {1} + \dots + a _ {n} \mu _ {n} \geq 0. $$

If for any such polynomial $ \Phi ( P) > 0 $, then the sequence is called strictly positive. For the sequence $ \mu _ {0} , \mu _ {1} \dots $ in $ [ a, b] $ to be positive, the existence of an increasing function $ g $ on $ [ a, b] $ for which

$$ \tag{1 } \int\limits _ { a } ^ { b } x ^ {n} dg( x) = \mu _ {n} ,\ \ n = 0, 1 \dots $$

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 $ \{ \mu _ {n} \} $ of real numbers there is a positive Borel measure $ \mu $ on $ \mathbf R $ such that $ \mu _ {n} = \int _ {\mathbf R } x ^ {n} d \mu ( x) $ 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