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

with radius of convergence , , the exponents are such that for an infinite set of values of :

where is a fixed positive number, then the sequence of partial sums of orders ,

converges uniformly in a sufficiently small neighbourhood of each point of the circle on which the sum of the series for is regular.

2) If

then the sequence converges uniformly in any closed bounded part of the domain of existence of .

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

with radius of convergence , , has a subsequence of partial sums that is uniformly convergent in some neighbourhood of , , then this power series can be represented as the sum of a series with radius of convergence greater than and a lacunary power series:

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)

References

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