Criterion for boundedness of power series

Importance: Low ✭
Author(s): Rüdinger, Andreas
Subject: Analysis
Recomm. for undergrads: yes
Posted by: andreasruedinger
on: May 9th, 2009

\begin{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 $x \in \mathbb{R}$. \end{question}

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: \begin{itemize} \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. \end{itemize}

A necessary condition

It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n.

What you have then is a

What you have then is a polynomial, and any nonconstant polynomial function is unbounded.

Re: A necessary condition

I posted the above comment anonymously, but now I have created an account. "It seems the sum would be bounded if there are only finitely many non-zero a sub n; it is not apparent to me that a sub 0 be the only non-zero a sub n."

harder than that

Look at sin(x)=x-x^3/6+x^5/120-....

JPB

sin x = cos(pi/2 - x)

The sine function is in the class mentioned.

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