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Convergence of a certain subsequence of partial sums of a series in a domain that is larger than the domain of convergence of the series. The following theorems on over-convergence hold:
 
Convergence of a certain subsequence of partial sums of a series in a domain that is larger than the domain of convergence of the series. The following theorems on over-convergence hold:
  
 
1) If for a power series
 
1) If for a power series
  
<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/o/o070/o070650/o0706501.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \
 +
\sum _ { n= } 1 ^  \infty 
 +
a _ {n} z ^ {\lambda _ {n} }
 +
$$
  
with radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706502.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706503.png" />, the exponents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706504.png" /> are such that for an infinite set of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706505.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706506.png" />:
+
with radius of convergence $  R $,
 +
$  0 < R < \infty $,  
 +
the exponents $  \lambda _ {n} $
 +
are such that for an infinite set of values $  n _  \nu  $
 +
of $  n $:
  
<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/o/o070/o070650/o0706507.png" /></td> </tr></table>
+
$$
 +
\lambda _ {n _  \nu  + 1 } - \lambda _ {n _  \nu  }  > \
 +
\theta \lambda _ {n _  \nu  } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706508.png" /> is a fixed positive number, then the sequence of partial sums of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o0706509.png" />,
+
where $  \theta $
 +
is a fixed positive number, then the sequence of partial sums of orders $  n _  \nu  $,
  
<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/o/o070/o070650/o07065010.png" /></td> </tr></table>
+
$$
 +
S _ { n _  \nu  } ( z)  = \
 +
\sum _ { m= } 1 ^ { {n _ \nu } }
 +
a _ {m} z ^ {\lambda _ {m} } ,\ \
 +
\nu = 1 , 2 \dots
 +
$$
  
converges uniformly in a sufficiently small neighbourhood of each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065011.png" /> of the circle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065012.png" /> on which the sum of the series for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065013.png" /> is regular.
+
converges uniformly in a sufficiently small neighbourhood of each point $  z _ {0} $
 +
of the circle $  | z | = R $
 +
on which the sum of the series for $  f ( z) $
 +
is regular.
  
 
2) If
 
2) If
  
<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/o/o070/o070650/o07065014.png" /></td> </tr></table>
+
$$
 +
\lambda _ {n _  \nu  + 1 } - \lambda _ {n _  \nu  }  > \
 +
\theta _  \nu  \lambda _ {n _  \nu  } ,\ \
 +
\lim\limits _ {\nu \rightarrow \infty }  \theta _  \nu  = + \infty ,
 +
$$
  
then the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065015.png" /> converges uniformly in any closed bounded part of the domain of existence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065016.png" />.
+
then the sequence $  \{ S _ {n _  \nu  } ( z) \} $
 +
converges uniformly in any closed bounded part of the domain of existence of $  f ( z) $.
  
 
The following theorem also holds (the converse of 1)): If a power series
 
The following theorem also holds (the converse of 1)): If a power series
  
<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/o/o070/o070650/o07065017.png" /></td> </tr></table>
+
$$
 +
f ( z)  = \
 +
\sum _ { n= } 0 ^  \infty 
 +
a _ {n} z  ^ {n}
 +
$$
  
with radius of convergence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065019.png" />, has a subsequence of partial sums that is uniformly convergent in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065021.png" />, then this power series can be represented as the sum of a series with radius of convergence greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/o/o070/o070650/o07065022.png" /> and a [[Lacunary power series|lacunary power series]]:
+
with radius of convergence $  R $,
 +
$  0 < R < \infty $,  
 +
has a subsequence of partial sums that is uniformly convergent in some neighbourhood of $  z _ {0} $,  
 +
$  | z _ {0} | \geq  R $,  
 +
then this power series can be represented as the sum of a series with radius of convergence greater than $  R $
 +
and a [[Lacunary power series|lacunary power series]]:
  
<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/o/o070/o070650/o07065023.png" /></td> </tr></table>
+
$$
 +
\sum _ { n= } 1 ^  \infty  d _ {n} z ^ {\lambda _ {n} } ,\ \
 +
\lambda _ {n _ {k}  + 1 } - \lambda _ {n _ {k}  }  > \
 +
\theta \lambda _ {n _ {k}  } ,\ \
 +
k = 1 , 2 ,\dots ; \ \
 +
\theta > 0.
 +
$$
  
 
The first theorem is true for many other series, in particular for [[Dirichlet series|Dirichlet series]].
 
The first theorem is true for many other series, in particular for [[Dirichlet series|Dirichlet series]].
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Bieberbach,  "Analytische Fortsetzung" , Springer  (1955)  pp. Sect. 3</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  G.M. Goluzin,  "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.F. Leont'ev,  "Exponential series" , Moscow  (1976)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. [L. Il'ev] Ilieff,  "Analytische Nichtfortsetzbarkeit und Überkonvergenz einiger Klassen von Potenzreihen" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L. [L. Il'ev] Ilieff,  "Analytische Nichtfortsetzbarkeit und Überkonvergenz einiger Klassen von Potenzreihen" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR></table>

Revision as of 08:04, 6 June 2020


Convergence of a certain subsequence of partial sums of a series in a domain that is larger than the domain of convergence of the series. The following theorems on over-convergence hold:

1) If for a power series

$$ f ( z) = \ \sum _ { n= } 1 ^ \infty a _ {n} z ^ {\lambda _ {n} } $$

with radius of convergence $ R $, $ 0 < R < \infty $, the exponents $ \lambda _ {n} $ are such that for an infinite set of values $ n _ \nu $ of $ n $:

$$ \lambda _ {n _ \nu + 1 } - \lambda _ {n _ \nu } > \ \theta \lambda _ {n _ \nu } , $$

where $ \theta $ is a fixed positive number, then the sequence of partial sums of orders $ n _ \nu $,

$$ S _ { n _ \nu } ( z) = \ \sum _ { m= } 1 ^ { {n _ \nu } } a _ {m} z ^ {\lambda _ {m} } ,\ \ \nu = 1 , 2 \dots $$

converges uniformly in a sufficiently small neighbourhood of each point $ z _ {0} $ of the circle $ | z | = R $ on which the sum of the series for $ f ( z) $ is regular.

2) If

$$ \lambda _ {n _ \nu + 1 } - \lambda _ {n _ \nu } > \ \theta _ \nu \lambda _ {n _ \nu } ,\ \ \lim\limits _ {\nu \rightarrow \infty } \theta _ \nu = + \infty , $$

then the sequence $ \{ S _ {n _ \nu } ( z) \} $ converges uniformly in any closed bounded part of the domain of existence of $ f ( z) $.

The following theorem also holds (the converse of 1)): If a power series

$$ f ( z) = \ \sum _ { n= } 0 ^ \infty a _ {n} z ^ {n} $$

with radius of convergence $ R $, $ 0 < R < \infty $, has a subsequence of partial sums that is uniformly convergent in some neighbourhood of $ z _ {0} $, $ | z _ {0} | \geq R $, then this power series can be represented as the sum of a series with radius of convergence greater than $ R $ and a lacunary power series:

$$ \sum _ { n= } 1 ^ \infty d _ {n} z ^ {\lambda _ {n} } ,\ \ \lambda _ {n _ {k} + 1 } - \lambda _ {n _ {k} } > \ \theta \lambda _ {n _ {k} } ,\ \ k = 1 , 2 ,\dots ; \ \ \theta > 0. $$

The first theorem is true for many other series, in particular for Dirichlet series.

References

[1] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3
[2] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)
[3] A.F. Leont'ev, "Exponential series" , Moscow (1976) (In Russian)

Comments

References

[a1] L. [L. Il'ev] Ilieff, "Analytische Nichtfortsetzbarkeit und Überkonvergenz einiger Klassen von Potenzreihen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
How to Cite This Entry:
Over-convergence. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Over-convergence&oldid=18212
This article was adapted from an original article by A.F. Leont'ev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article