Difference between revisions of "Lacunary power series"
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| − | + | \begin{equation}\label{lac} f(z)=\sum_{k=1}^\infty a_kz^{\lambda_k}\end{equation} | |
| − | with gaps (lacunas), so that the exponents | + | with gaps (lacunas), so that the exponents $\lambda_1,\lambda_2,\ldots,$ do not run through all the natural numbers. Depending on the properties of the sequence $\{\lambda_k\}$ one obtains many properties of the series \ref{lac}. Thus, if |
| − | + | $$\lambda_{k+1}-\lambda_k>\theta\lambda_k,\quad k=0,1,\ldots,\quad \theta>0,$$ | |
| − | and the series | + | and the series \ref{lac} converges in the disc $\lvert z\rvert<R$, $0<R<\infty$, then all points of the circle $\lvert z\rvert=R$ are singular for $f(z)$ ([[Hadamard theorem#Hadamard's gap theorem|Hadamard's gap theorem]]). A strengthening of this theorem is Fabry's gap theorem (cf. [[Fabry theorem|Fabry theorem]]). If the lower density |
| − | + | $$\liminf_{k\to\infty}\frac{k}{\lambda_k}=0,$$ | |
| − | then | + | then $f(z)$ is a single-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also [[Over-convergence|Over-convergence]]. |
====References==== | ====References==== | ||
| − | + | {| | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Bi}}||valign="top"| L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Di}}||valign="top"| P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957) | |
| − | + | |- | |
| − | + | |valign="top"|{{Ref|Ti}}||valign="top"| E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) | |
| − | + | |- | |
| − | + | |} | |
Revision as of 06:16, 22 April 2012
A series
\begin{equation}\label{lac} f(z)=\sum_{k=1}^\infty a_kz^{\lambda_k}\end{equation}
with gaps (lacunas), so that the exponents $\lambda_1,\lambda_2,\ldots,$ do not run through all the natural numbers. Depending on the properties of the sequence $\{\lambda_k\}$ one obtains many properties of the series \ref{lac}. Thus, if
$$\lambda_{k+1}-\lambda_k>\theta\lambda_k,\quad k=0,1,\ldots,\quad \theta>0,$$
and the series \ref{lac} converges in the disc $\lvert z\rvert<R$, $0<R<\infty$, then all points of the circle $\lvert z\rvert=R$ are singular for $f(z)$ (Hadamard's gap theorem). A strengthening of this theorem is Fabry's gap theorem (cf. Fabry theorem). If the lower density
$$\liminf_{k\to\infty}\frac{k}{\lambda_k}=0,$$
then $f(z)$ is a single-valued analytic function with simply-connected domain of existence (Pólya's theorem). See also Over-convergence.
References
| [Bi] | L. Bieberbach, "Analytische Fortsetzung" , Springer (1955) pp. Sect. 3 |
| [Di] | P. Dienes, "The Taylor series" , Oxford Univ. Press & Dover (1957) |
| [Ti] | E.C. Titchmarsh, "The theory of functions" , Oxford Univ. Press (1979) |
How to Cite This Entry:
Lacunary power series. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lacunary_power_series&oldid=24997
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