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Lacunary power series

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A series

(*)

with gaps (lacunas), so that the exponents do not run through all the natural numbers. Depending on the properties of the sequence one obtains many properties of the series (*). Thus, if

and the series (*) converges in the disc , , then all points of the circle are singular for (Hadamard's gap theorem). A strengthening of this theorem is Fabry's gap theorem (cf. Fabry theorem). If the lower density

then is a single-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also Over-convergence.

References

[1] L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3


Comments

References

[a1] E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979)
[a2] P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957)
How to Cite This Entry:
Lacunary power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lacunary_power_series&oldid=24997
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