Over-convergence

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)
[a1]	L. [L. Il'ev] Ilieff, "Analytische Nichtfortsetzbarkeit und Überkonvergenz einiger Klassen von Potenzreihen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)