View source for Over-convergence

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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:

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|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|Dirichlet series]].

====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>

====Comments====

====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>