Criterion for Boundedness of Power Series

Tue, 26 May 2026 00:00:00 +0000

Introduction & Problem Statement #

Power series constitute one of the most ubiquitous objects in analysis. A power series $\sum_{n=0}^{\infty}a_n x^n$ with infinite radius of convergence defines a real-entire function $f:\mathbb{R}\to\mathbb{R}$. Whereas the question of convergence is completely settled by Cauchy–Hadamard theory, the question of boundedness of the sum function is far subtler and, as of this writing, remains open.

Question 1 (Rüdinger, 2009)

Let $(a_n) _{n\ge 0}$ be a sequence of real numbers such that the power series $\sum _{n=0}^{\infty}a_n x^n$ converges for every $x\in\mathbb{R}$, thereby defining a smooth function $f:\mathbb{R}\to\mathbb{R}$. Give a necessary and sufficient criterion on $(a_n)$ for $f$ to be bounded on $\mathbb{R}$.

The problem is rated low importance on the Open Problem Garden and is recommended as accessible to undergraduates; nevertheless, a complete answer appears to be unknown.

Motivating examples.

Function	Power series	Bounded?
$\cos x$	$\displaystyle\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k)!}x^{2k}$	$\|\cos x\|\le 1$
$\sin x$	$\displaystyle\sum_{k=0}^{\infty}\frac{(-1)^k}{(2k+1)!}x^{2k+1}$	$\|\sin x\|\le 1$
$e^x$	$\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}$	$e^x\to+\infty$
$p(x)=a_0+\cdots+a_Nx^N,\ N\ge 1$	(polynomial)	unbounded

Background & Prerequisites #

This section collects the core mathematical tools needed to engage seriously with Question 1.

Power Series and Entire Functions #

Definition 1 (Power Series & Radius of Convergence)

A power series centred at the origin is a formal series $\sum_{n=0}^{\infty}a_n x^n$ with $a_n\in\mathbb{R}$. Its radius of convergence is $$ R = \frac{1}{\limsup_{n\to\infty}|a_n|^{1/n}} \in [0,+\infty]. $$

Throughout this note we always assume $R=+\infty$, i.e., $\limsup_{n\to\infty}|a_n|^{1/n}=0$.

Definition 2 (Entire Function)

A function $f:\mathbb{C}\to\mathbb{C}$ is called entire if it is holomorphic on all of $\mathbb{C}$. Every power series with $R=+\infty$ defines a real-entire function, and by the identity theorem its complex extension is entire.

Theorem 1 (Cauchy–Hadamard)

The radius of convergence of $\sum a_n z^n$ equals $$ R = \Bigl(\limsup_{n\to\infty}|a_n|^{1/n}\Bigr)^{-1}. $$

Remark 1

The condition $R=+\infty$ is equivalent to $a_n = O(r^n/n!)$ for every $r>0$, i.e., the coefficients decay faster than any geometric sequence. This is the Paley–Wiener type condition for entire functions of order $1$.

Order and Type of Entire Functions #

Definition 3 (Order and Type)

The order of an entire function $f$ is $$ \rho = \limsup_{r\to\infty}\frac{\log\log M(r)}{\log r}, \qquad M(r)=\max_{|z|=r}|f(z)|. $$ The type $\sigma$ (for $0<\rho<\infty$) is $$ \sigma = \limsup_{r\to\infty}\frac{\log M(r)}{r^{\rho}}. $$

A bounded complex entire function has order $\rho=0$ (by Liouville’s theorem it must be constant), while a bounded real-valued entire function can be non-constant. Boundedness is therefore a genuinely real-variable phenomenon.

Liouville’s Theorem and Its Limitations #

Theorem 2 (Liouville)

Every bounded entire function $f:\mathbb{C}\to\mathbb{C}$ is constant.

Remark 2 (Why Liouville does not solve the problem)

Question 1 concerns real-valued functions $f:\mathbb{R}\to\mathbb{R}$. A function may be bounded on $\mathbb{R}$ while its complex extension is unbounded. For instance, $\cos z$ satisfies $|\cos z|\to\infty$ along the imaginary axis (since $\cos(iy)=\cosh y\to+\infty$). Liouville’s theorem therefore does not apply, and the problem is genuinely non-trivial.

Algebraic Structure of the Relevant Function Space #

Definition 4 (Space of Bounded Power Series)

Let $\mathcal{B}$ denote the set of all functions $f:\mathbb{R}\to\mathbb{R}$ that can be represented as a convergent power series $\sum_{n\ge 0}a_n x^n$ (with $R=+\infty$) and that are bounded on $\mathbb{R}$.

Proposition 1, Algebraic Properties of $\mathcal{B}$ (Rüdinger, 2009)

$\mathcal{B}$ is a linear subspace of $C^\infty(\mathbb{R})$: if $f,g\in\mathcal{B}$ and $\lambda\in\mathbb{R}$ then $f+\lambda g\in\mathcal{B}$.
$\mathcal{B}$ is closed under pointwise multiplication: if $f,g\in\mathcal{B}$ then $fg\in\mathcal{B}$.
$\mathcal{B}$ contains all functions of the form $c\cos(h(x))$, where $c\in\mathbb{R}$ and $h:\mathbb{R}\to\mathbb{R}$ is any entire function.

Remark 3

Part (3) follows from $\cos(h(x)) = \operatorname{Re}(e^{ih(x)})$ together with $|\cos(h(x))|\le 1$. The class is strictly larger than ${c\cos(bx):c,b\in\mathbb{R}}$; for example, $\cos(x^3-x)\in\mathcal{B}$.

Known Partial Results #

Necessary Conditions #

Proposition 2, Necessary Condition for Boundedness (Rüdinger, 2009)

Suppose $f(x)=\sum_{n=0}^{\infty}a_n x^n$ is bounded on $\mathbb{R}$. Then either:

$a_0$ is the only non-zero coefficient (i.e., $f$ is the constant function $f\equiv a_0$), or
there are infinitely many indices $n$ with $a_n\neq 0$, and the signs of the non-zero $a_n$ change infinitely often.

Remark 4

The sign-change condition is necessary: if the non-zero coefficients are eventually of one sign, the dominant-term comparison shows $f(x)\to\pm\infty$ as $x\to+\infty$ or $x\to-\infty$.

Corollary 1

Every non-constant polynomial is unbounded on $\mathbb{R}$.

Proof.

A polynomial has only finitely many non-zero coefficients. By Proposition 2 (1), the only bounded polynomial is the constant function. Any non-constant polynomial satisfies $|p(x)|\to\infty$ as $|x|\to\infty$.

The Sign-Change Condition Is Not Sufficient #

The condition of Proposition 2 is not sufficient, as the following examples show.

Example 1

Consider the geometric series $$ f(x) = \sum_{n=0}^{\infty}(-1)^n x^{2n} = \frac{1}{1+x^2}, \qquad |x|<1. $$ The coefficients alternate in sign, yet $R=1\neq+\infty$. One must first require $R=+\infty$ before the sign-change condition becomes meaningful.

For a subtler case with $R=+\infty$: take $a_n=(-1)^n/n!$, so $$ f(x) = \sum_{n=0}^{\infty}\frac{(-1)^n}{n!}x^n = e^{-x}. $$ The signs alternate, yet $e^{-x}\to+\infty$ as $x\to-\infty$.

Remark 5

The $e^{-x}$ example reveals the key gap: sign alternation of the coefficients does not prevent the function from growing in one direction, because the series for $e^{-x}$ reconstructs exponential growth in the negative half-line. A complete criterion must capture cancellation in both directions.

Connections to Entire Function Theory #

Theorem 3 (Borel–Carathéodory)

Let $f$ be holomorphic in $|z|\le R$. Then for $0<r<R$, $$ M(r) \le \frac{2r}{R-r}\sup_{|z|=R}\operatorname{Re}f(z) + \frac{R+r}{R-r},|f(0)|. $$

Remark 6

Borel–Carathéodory shows that the real part of a complex-valued entire function controls its modulus. For a real-valued function on $\mathbb{R}$ the analogous control is more delicate, since we only observe the function on a line, not on a disk.

Theorem 4 (Hadamard Factorisation)

Every entire function of finite order $\rho$ can be written as $$ f(z) = z^m e^{g(z)}\prod_{n=1}^{\infty} E_p!\left(\frac{z}{z_n}\right), $$ where $m\ge 0$, $p=\lfloor\rho\rfloor$, $g$ is a polynomial of degree $\le\rho$, and the $E_p$ are Weierstrass elementary factors.

Remark 7

A bounded real entire function of infinite order (if one exists) would not be directly covered by the Hadamard factorisation. Understanding the zero set and the exponential factor in $e^{g(z)}$ may be key to classifying all $f\in\mathcal{B}$.

The Open Sub-Question on the Generators of $\mathcal{B}$ #

Question 2 (Rüdinger, 2009)

Does $\mathcal{B}$ consist precisely of functions of the form $c\cos(h(x))$ and their linear combinations and products, where $h:\mathbb{R}\to\mathbb{R}$ is entire and $c\in\mathbb{R}$?

A positive answer would give an implicit characterisation via algebraic generators. A negative answer would require producing a bounded entire function on $\mathbb{R}$ that does not lie in the $\mathbb{R}$-algebra generated by ${\cos\circ, h : h\text{ entire}}$.

Remark 8

By Proposition 1 (3), every $c\cos(h(x))$ belongs to $\mathcal{B}$, and $\mathcal{B}$ is an algebra, so all products and sums remain in $\mathcal{B}$. What is unknown is whether every element of $\mathcal{B}$ arises this way. Note that $\sin x = \cos(x-\pi/2) \in \mathcal{B}$, so sine is already covered.

Research Directions and Conjectures #

Direction 1: Coefficient Growth Rate #

A promising approach is to examine the rate of decay of $|a_n|$, not just the sign pattern.

Question 3

Is there a decay condition on $|a_n|$, combined with the sign-change condition, that gives a sufficient criterion for $f\in\mathcal{B}$?

Approach. The Cauchy estimates give $|a_n| = |f^{(n)}(0)|/n!\le M(r)/r^n$ for all $r>0$. If $f\in\mathcal{B}$ with $|f|\le B$, the bound $|a_n|\le B/r^n$ holds for every $r>0$, but this recovers only the $R=+\infty$ condition. Is there a sharper constraint?

Direction 2: Fourier-Analytic Approach #

Every $f\in L^\infty(\mathbb{R})\cap L^2(\mathbb{R})$ possesses a square-integrable Fourier transform. If $f$ is also entire, Paley–Wiener forces the transform to be compactly supported. However, a generic $f\in\mathcal{B}$ may not lie in $L^2$ (e.g., $\cos x\notin L^2(\mathbb{R})$).

Question 4

Can the Fourier theory for tempered distributions give a necessary and sufficient condition for $f\in\mathcal{B}$ in terms of the spectral support of $f$?

Direction 3: Differential Equation Characterisation #

Bounded entire functions often arise as solutions to ODEs. For instance $y’’+y=0$ has bounded solutions $A\cos x + B\sin x$. More generally, $y’’+\omega(x)y=0$ with $\omega$ entire and bounded can produce bounded solutions.

Question 5

Characterise those linear differential operators $L$ with entire coefficients whose full solution space lies within $\mathcal{B}$.

Direction 4: Even/Odd Decomposition and Reduction #

Every $f\in\mathcal{B}$ splits as $f=f_e+f_o$ where $$ f_e(x)=\tfrac{1}{2}(f(x)+f(-x))=\sum_{k\ge 0}a_{2k}x^{2k} \quad\text{and}\quad f_o(x)=\tfrac{1}{2}(f(x)-f(-x))=\sum_{k\ge 0}a_{2k+1}x^{2k+1}. $$ Since $f_e(x)=g(x^2)$ for the entire function $g(t)=\sum_{k\ge 0}a_{2k}t^k$, boundedness of $f_e$ reduces to: is $g$ bounded on $[0,+\infty)$? This reduction may make the even and odd parts easier to study separately.

Direction 5: Polynomial Approximation and Numerics #

Question 6

If the partial sums $S_N(x)=\sum_{n=0}^{N}a_n x^n$ are uniformly bounded on growing intervals $[-R_N,R_N]$ (with $R_N\to\infty$), does it follow that $f\in\mathcal{B}$? Conversely, if $f\in\mathcal{B}$, how fast must $R_N$ grow relative to $N$ for the bound to hold?

Summary of Open Problems #

#	Statement
Q1	Give a necessary and sufficient condition on $(a_n)$ for $f=\sum a_n x^n$ to be bounded on $\mathbb{R}$.
Q2	Is $\mathcal{B}$ generated (as an algebra) precisely by ${c\cos(h(x)):h\text{ entire}}$?
Q3	Does a sharper decay condition on $
Q4	Can spectral-support (Paley–Wiener / distribution) theory characterise $\mathcal{B}$?
Q5	Which linear ODEs with entire coefficients have solution space $\subseteq\mathcal{B}$?
Q6	What is the precise relationship between truncation bounds on $[-R_N,R_N]$ and $f\in\mathcal{B}$?

References #

Ahlfors, L. V. (1979). Complex Analysis, 3rd ed. McGraw-Hill.
Boas, R. P. (1954). Entire Functions. Academic Press.
Conway, J. B. (1978). Functions of One Complex Variable, 2nd ed. Springer.
Levin, B. Ya. (1996). Lectures on Entire Functions. AMS Translations of Mathematical Monographs, vol. 150.
Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill.
Rudin, W. (1987). Real and Complex Analysis, 3rd ed. McGraw-Hill.
Rüdinger, A. (2009). Criterion for boundedness of power series. Open Problem Garden. http://www.openproblemgarden.org/op/criterion_for_boundedness_of_power_series
Stein, E. M. and Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University Press.
Stein, E. M. and Shakarchi, R. (2010). Complex Analysis. Princeton University Press.
Titchmarsh, E. C. (1939). The Theory of Functions, 2nd ed. Oxford University Press.

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