Importance: Low ✭
Author(s): Rüdinger, Andreas
Subject: Analysis
Recomm. for undergrads: yes
Posted by: andreasruedinger
on: May 9th, 2009
Question   Give a necessary and sufficient criterion for the sequence $ (a_n) $ so that the power series $ \sum_{n=0}^{\infty} a_n x^n $ is bounded for all $ x \in \mathbb{R} $.

Consider a power series $ \sum_{n=0}^{\infty} a_n x^n $ that is convergent for all $ x \in {\mathbb R}  $, thus defining a function $ f: {\mathbb R} \to {\mathbb R} $. Are there criteria to decide whether $ f $ is bounded (which e.g. is the case for the series with $ a_n = (-1)^k/(2k)! $ for $ n = 2k $ and $ a_n = 0 $ for n odd)? Some general remarks:

    \item A necessary condition for $ \sum_n a_n x^n $ to be bounded is that $ a_0 $ is the only non-zero $ a_n $ or there are infinitely many non-zero $ a_n $'s which change sign infinitely many times. \item Changing a finite set of $ a_n $'s (except $ a_0 $) does leave the subspace of bounded power series. \item The subspace of bounded power series is "large" in the sense that it is both a linear subspace (closed under sums and scalar multiples) and an algebra (closed under products). It includes all functions of the form $ a \cos( f(x)) $, where $ f $ is any entire function $ \mathbb{R} \to \mathbb{R} $. The question whether the subspace of bounded power series contains only these functions seems to be open.


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