View source for Positive sequence

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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|Moment problem]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H.J. Landau (ed.) , ''Moments in mathematics'' , Amer. Math. Soc.  (1987)  pp. 56ff</TD></TR></table>