# 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)