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A sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739601.png" /> of real numbers in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739602.png" /> such that for any polynomial
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739603.png" /></td> </tr></table>
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that is not identically zero and is not negative on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739604.png" /> the expression
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A sequence  $  \mu _ {0} , \mu _ {1} \dots $
 +
of real numbers in the interval  $  [ a, b] $
 +
such that for any polynomial
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739605.png" /></td> </tr></table>
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$$
 +
P( x)  = a _ {0} + a _ {1} x + \dots + a _ {n} x  ^ {n}
 +
$$
  
If for any such polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739606.png" />, then the sequence is called strictly positive. For the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739607.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739608.png" /> to be positive, the existence of an increasing function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p0739609.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396010.png" /> for which
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that is not identically zero and is not negative on  $  [ a, b] $
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the expression
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396011.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
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$$
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\Phi ( P)  = a _ {0} \mu _ {0} + a _ {1} \mu _ {1} + \dots + a _ {n} \mu _ {n}  \geq
 +
0.
 +
$$
  
is necessary and sufficient.
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If for any such polynomial  $  \Phi ( P) > 0 $,
 +
then the sequence is called strictly positive. For the sequence  $  \mu _ {0} , \mu _ {1} \dots $
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in  $  [ a, b] $
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to be positive, the existence of an increasing function  $  g $
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on  $  [ a, b] $
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for which
  
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$$ \tag{1 }
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\int\limits _ { a } ^ { b }  x  ^ {n}  dg( x)  =  \mu _ {n} ,\ \
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n = 0, 1 \dots
 +
$$
  
 +
is necessary and sufficient.
  
 
====Comments====
 
====Comments====
A (strictly) negative sequence can be similarly defined and has a similar property. The problem of deciding whether for a given sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396012.png" /> of real numbers there is a positive Borel measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396013.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396014.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073960/p07396015.png" /> is known as the Hamburger moment problem. The condition (1) is a moment condition, cf. [[Moment problem|Moment problem]].
+
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====
 
====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>
 
<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>

Latest revision as of 08:07, 6 June 2020


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
How to Cite This Entry:
Positive sequence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_sequence&oldid=48256
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article