1 Introduction
Tauberian theorems constitute a major branch of summability theory in mathematical analysis. They provide conditions under which a sequence or function that is summable by a weaker method (e.g., Cesàro or Abel summation) actually converges in the ordinary sense. The typical structure pairs an *Abelian theorem* (which asserts that ordinary convergence implies summability by a given method) with a *Tauberian theorem* (which gives a converse under an additional restriction, known as a *Tauberian condition*). These results have profound applications in analytic number theory, harmonic analysis, and probability.
1.1 Historical background
The first Tauberian theorem was proved by the Austrian mathematician Alfred Tauber in 1897. Tauber considered Cesàro summability of order 1 (the method of arithmetic means) and showed that if a series is Cesàro summable and its terms satisfy a certain boundedness condition (namely, $n a_n = o(n)$ for the sequence of partial sums), then the series converges in the ordinary sense. This sparked immediate interest. G. H. Hardy and J. E. Littlewood extended the theory to Abel summability in the 1910s, introducing the famous one-sided Tauberian condition $a_n \ge -C$ for some constant $C$. In the 1930s, Norbert Wiener revolutionized the field by placing Tauberian theorems in the context of Fourier analysis and Banach algebras, leading to a general theory that unified many disparate results. Wiener’s work also provided a new proof of the Prime Number Theorem.
1.2 Basic definitions and notation
Throughout the article, let $\{s_n\}_{n \ge 0}$ denote a sequence of real or complex numbers, often interpreted as the partial sums of a series $\sum_{k=0}^\infty a_k$ (so $s_n = \sum_{k=0}^n a_k$). For integrals, we consider functions $F(x)$ for $x \ge 0$, with the integral $\int_0^\infty f(t)\,dt$ in the Riemann or Lebesgue sense.
1.2.1 Summability methods overview
A *summability method* is a procedure that assigns a limit to a sequence or series when the ordinary limit does not exist. Common methods include:
- Cesàro summability (C,1): The sequence $\{s_n\}$ is Cesàro summable to $L$ if $\lim_{n\to\infty} \frac{1}{n+1}\sum_{k=0}^n s_k = L$.
| - Abel summability: The series $\sum a_k$ is Abel summable to $L$ if $\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = L$ (provided the power series converges for $ | x | <1$). |
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- Matrix summability: A general class where the sequence is transformed by an infinite matrix (e.g., the Cesàro matrix).
1.2.2 Ordinary convergence vs. summability
Ordinary convergence is the strongest standard notion: $\lim_{n\to\infty} s_n = L$. A summability method is *regular* if it assigns the same limit as ordinary convergence whenever the latter exists. Most classical methods are regular. The goal of a Tauberian theorem is to reverse the direction: from summability back to convergence, under a Tauberian condition that restricts the “oscillatory” behavior of the sequence or function.
2 Classical Tauberian theorems
Classical results focus on the Cesàro and Abel methods, establishing precise conditions under which the converse holds.
2.1 Tauber's original theorem for Cesàro summability
Tauber’s 1897 theorem was the first result of its kind.
2.1.1 Statement and proof sketch
Let $\{s_n\}$ be a sequence of real numbers. If
- $\lim_{n\to\infty} \frac{1}{n+1}\sum_{k=0}^n s_k = L$ (Cesàro summability), and
- $s_n - s_{n-1} = o(1/n)$ (i.e., $n(s_n - s_{n-1}) \to 0$),
then $\lim_{n\to\infty} s_n = L$. The proof uses the identity $s_n = (n+1)\sigma_n - n\sigma_{n-1}$ where $\sigma_n$ are the Cesàro means, and estimates the difference $s_n - \sigma_n$.
2.1.2 Tauberian condition for Cesàro means
The condition $n a_n \to 0$ (where $a_n = s_n - s_{n-1}$) is often called the *little-o* Tauberian condition for Cesàro summability. It is sharp: without it, Cesàro summability does not imply convergence (e.g., the series $1-1+1-1+\cdots$ is Cesàro summable to $1/2$ but does not converge). Variants use boundedness ($n a_n = O(1)$) for Cesàro methods of higher order.
2.2 Littlewood's theorem for Abel summability
In 1910, J. E. Littlewood proved a landmark theorem for Abel summability.
2.2.1 The one-sided Tauberian condition
Let $\sum_{k=0}^\infty a_k$ be a series with real terms. If
- $\lim_{x\to 1^-} \sum_{k=0}^\infty a_k x^k = L$ (Abel summable), and
- $a_k \ge -C$ for some constant $C$ (or more generally, the partial sums are bounded below),
then the series converges to $L$ in the ordinary sense. The condition $a_k \ge -C$ is called the *one-sided Tauberian condition*. It cannot be replaced by $a_k \le C$ (consider the series with $a_k = (-1)^k$, which is Abel summable to $1/2$ but fails convergence).
2.2.2 Relationship with Cesàro methods
Hardy and Littlewood showed that Abel summability is stronger than Cesàro summability of any order. More precisely, (C,1)-summability implies Abel summability (Abelian theorem), and the converse holds under a suitable Tauberian condition. For example, if a series is Abel summable and its terms satisfy $a_k = O(1/k)$, then it is Cesàro summable (and hence convergent if further conditions hold). The one-sided condition suffices for Abel to Cesàro.
2.3 Hardy's theorem and related results
Hardy contributed several refinements, emphasizing uniformity and applications to power series.
2.3.1 Uniform Tauberian theorems
Hardy considered sequences depending on a parameter and proved conditions under which the convergence is uniform. For instance, if a family of series $\sum a_n(t)$ is Abel summable uniformly in $t$ and each satisfies a uniform one-sided condition, then the ordinary convergence is also uniform. Such results are crucial in harmonic analysis.
2.3.2 Application to power series
If a power series $\sum_{n=0}^\infty a_n z^n$ has radius of convergence $1$ and its boundary behavior is Abel summable at a point $z_0$ on the unit circle, then under the one-sided condition on the coefficients, the series converges at $z_0$. This connects Tauberian theory with the Fatou–Riesz theorem.
3 Wiener's general Tauberian theory
Norbert Wiener developed a sweeping generalization of Tauberian theorems using the Fourier transform and the theory of Banach algebras.
3.1 Wiener's theorem for the Fourier transform
Wiener’s approach recasts the Tauberian condition in terms of convolution and the non-vanishing of a Fourier transform.
3.1.1 The convolution kernel approach
Consider a function $f(t)$ on $\mathbb{R}$ and a kernel $K(t)$ satisfying $\int_{-\infty}^{\infty} K(t)\,dt = 1$. The *Abelian* direction says: if $\lim_{t\to\infty} f(t) = L$, then $\lim_{x\to\infty} \int K_x(t) f(t)\,dt = L$, where $K_x(t) = x K(xt)$. The *Tauberian* direction asks: if the convolution averages converge to $L$ for a suitable family of kernels, does $f(t)$ converge to $L$? Wiener showed that the answer is affirmative provided the Fourier transform $\hat{K}(\xi)$ never vanishes, together with a Tauberian condition on $f$ (e.g., boundedness).
3.1.2 Translation-invariant Banach algebras
Wiener’s theory is naturally phrased in terms of the Banach algebra $L^1(\mathbb{R})$ under convolution. A key result is: if $f \in L^\infty(\mathbb{R})$ and for some $K \in L^1$ with $\hat{K}(\xi) \neq 0$ for all $\xi$, the convolution $(K * f)(x) \to L \int K$ as $x\to\infty$, then for any $G \in L^1$, $(G*f)(x) \to L \int G$. This reduces Tauberian questions to ideal theory: the closure of translations of $K$ spans the whole algebra.
3.2 The Tauberian condition in Wiener's framework
Wiener’s theorem imposes conditions on the function (e.g., boundedness) rather than on its discrete increments.
3.2.1 Non-vanishing of Fourier transforms
The essential requirement is that $\hat{K}(\xi) \neq 0$ for all $\xi \in \mathbb{R}$. If $\hat{K}$ vanishes at some point, one can construct counterexamples: the function $f(t) = e^{i\xi_0 t}$ is such that $K * f \equiv 0$ while $f$ does not converge. This non-vanishing condition is both necessary and sufficient for the class of bounded functions satisfying the convolution limit to force convergence.
3.2.2 Closure properties of ideals
In the Banach algebra $L^1(\mathbb{R})$, the set of functions $\{K_a : a \in \mathbb{R}\}$ where $K_a(t)=K(t-a)$ generates a closed ideal. Wiener’s Tauberian theorem is equivalent to the statement that the only closed ideal containing a function $K$ whose Fourier transform never vanishes is the whole algebra. This deep algebraic insight unified many earlier results.
3.3 Extensions to locally compact abelian groups
The theory extends naturally to more general groups, especially those relevant to number theory.
3.3.1 Abstract harmonic analysis setting
Let $G$ be a locally compact abelian group (e.g., $\mathbb{R}^n$, the circle group $\mathbb{T}$, or the additive group of $p$-adic numbers). A function $f \in L^\infty(G)$ is said to have a limit at infinity (via a net of translations) if for some $L$, $(K * f)$ converges to $L$ for a suitable kernel. The analogue of Wiener’s theorem holds when $\hat{K}$ never vanishes on the dual group $\hat{G}$.
3.3.2 Applications to number theory
In analytic number theory, one often works with the group $\mathbb{R}$ (for Dirichlet series) or the group of idèles. The Wiener–Ikehara theorem (a Tauberian theorem for Dirichlet series with nonnegative coefficients) is the key to proving the Prime Number Theorem. It applies the general theory to the kernel $K(t) = e^{-t}$ for $t\ge 0$, whose Fourier transform $\frac{1}{1+i\xi}$ never vanishes.
4 Modern developments and variations
Since Wiener, many variations have appeared, addressing more general summability methods, higher orders, and functional-analytic frameworks.
4.1 Tauberian theorems for matrix summability
A general summability method can be defined by a matrix $A = (a_{nk})$: the $A$-transform of $\{s_n\}$ is $t_n = \sum_{k=0}^\infty a_{nk} s_k$.
4.1.1 Regular matrices and Toeplitz conditions
| A matrix is *regular* if it maps convergent sequences to convergent sequences preserving the limit. The Silverman–Toeplitz theorem gives necessary and sufficient conditions: $\lim_n a_{nk} = 0$ for each $k$, $\lim_n \sum_k a_{nk} = 1$, and $\sum_k | a_{nk} | $ is uniformly bounded. Tauberian theorems for such matrices typically require that the sequence be “slowly oscillating” or satisfy a condition like $s_n - s_{n-1} = o(1)$. |
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4.1.2 Mercerian theorems
A *Mercerian theorem* is a Tauberian theorem for which the summability method itself implies convergence without any additional Tauberian condition—i.e., the method is “stronger” than ordinary convergence. For example, the Cesàro method of order $r>0$ is not Mercerian (the sequence $(-1)^n$ is Cesàro summable but not convergent), but certain weighted means can be. Mercerian theorems are important in functional analysis and operator theory.
4.2 High-order Tauberian theorems
Higher-order Cesàro and other methods require more complex conditions.
4.2.1 k-th order Cesàro means
The Cesàro method of order $k$ (denoted $C,k$) uses iterated averages. For example, $(C,2)$ summability of $\{s_n\}$ means the limit of $\frac{2}{(n+1)(n+2)}\sum_{j=0}^n (n-j+1)s_j$ exists. A Tauberian theorem for $(C,k)$ typically involves the condition $n^k a_n = o(1)$ (or boundedness) for the sequence of differences.
4.2.2 Borel and Euler summability
Borel summability and Euler summability are methods often used for divergent series in complex analysis. Borel summability of $\sum a_n$ means $\int_0^\infty e^{-t} \sum_{n=0}^\infty a_n t^n/n! \,dt$ converges. Tauberian theorems for Borel summability require that $a_n$ be “slowly varying” or satisfy a condition like $a_n = O(e^{c\sqrt{n}})$. These have applications in asymptotic expansions and quantum field theory.
4.3 Functional-analytic approach
Modern treatments often place Tauberian theorems in the language of Banach algebras and operator theory.
4.3.1 Spectral theory and Banach algebras
Wiener’s theorem is an early instance of a general phenomenon: in a commutative Banach algebra with unit, the invertibility of an element $a$ is equivalent to the non-vanishing of its Gelfand transform on the maximal ideal space. Tauberian conditions can be interpreted as requiring that the sequence belongs to the closure of a certain ideal.
4.3.2 The role of weighted norms
Weighted $L^p$ norms, with weights like $e^{c t}$, can extend Tauberian theorems to rapidly growing sequences. For example, the abscissa of convergence of a Dirichlet series is related to Tauberian conditions on its coefficients. The use of weighted norm inequalities allows for Tauberian theorems for the Laplace transform and related transforms, leading to results in the study of entire functions.
5 Applications
Tauberian theorems are indispensable in several fields, providing the bridge between analytic and elementary properties.
5.1 Analytic number theory
The most famous application is the Prime Number Theorem.
5.1.1 Prime number theorem (via Wiener–Ikehara)
The Wiener–Ikehara theorem states: if $F(s) = \sum_{n=1}^\infty a_n n^{-s}$ is a Dirichlet series with nonnegative coefficients $a_n$, convergent for $\mathrm{Re}\,s > 1$ and extendable to a meromorphic function on $\mathrm{Re}\,s \ge 1$ with only a simple pole at $s=1$ with residue $R$, then $\sum_{n\le x} a_n \sim R x$. Applying this to $a_n = \Lambda(n)$ (the von Mangoldt function) yields $\psi(x) \sim x$, which is equivalent to the Prime Number Theorem.
5.1.2 Dirichlet series and Tauberian constants
More refined Tauberian theorems give asymptotic expansions with error terms. For instance, if $F(s)$ has a pole of higher order, one obtains asymptotic formulas with polynomial factors, e.g., for sums of divisor functions. The constants involved are often expressed in terms of residues of $F(s)$.
5.2 Fourier analysis and summability of integrals
Tauberian theorems control the convergence of Fourier integrals and series.
5.2.1 Fourier inversion and Cesàro means
The classical Dirichlet kernel yields pointwise convergence of Fourier series only under stringent conditions. Cesàro (Fejér) means converge uniformly for continuous functions. A Tauberian theorem for Fourier series: if the Cesàro means of a function converge at a point, and the function’s partial sums satisfy a condition like bounded variation, then the Fourier series converges at that point.
5.2.2 Abelian and Tauberian theorems for integrals
Consider a function $f(t)$ and its Laplace transform $\mathcal{L}f(s) = \int_0^\infty e^{-st} f(t)\,dt$. The abelian direction: if $f(t)\to L$, then $\mathcal{L}f(s) \sim L/s$ as $s\to 0^+$. The Tauberian direction (Karamata’s theorem): if $\mathcal{L}f(s) \sim Cs^{-\rho}$ as $s\to 0^+$ and $f$ is nonnegative, then $\int_0^x f(t)\,dt \sim C x^\rho/\Gamma(\rho+1)$. This is a cornerstone of Tauberian theory for integrals.
5.3 Probability theory and stochastic processes
Tauberian theorems arise naturally in the study of limit theorems and tail behavior.
5.3.1 Renewal theory and Tauberian conditions
In renewal theory, the renewal function $U(t) = \sum_{n=0}^\infty \mathbb{P}(S_n \le t)$ often satisfies an integral equation whose Laplace transform is known. A Tauberian theorem applied to $U(t)$ gives the asymptotic behavior of the number of renewals. The Tauberian condition is typically the existence of a density or regular variation.
5.3.2 Heavy-tailed distributions
For heavy-tailed distributions, the tail probability $\bar{F}(x) = \mathbb{P}(X > x)$ decays like a power law. The asymptotic behavior of $\bar{F}(x)$ can be deduced from the behavior of the Laplace–Stieltjes transform near $0$ using Tauberian theorems. Conversely, if the transform has a certain asymptotic, the tail is regularly varying.
6 Related concepts and further reading
6.1 Abelian theorems (the converse direction)
Abelian theorems form the opposite direction: if a sequence converges, then it is summable by a given method. They are typically easier to prove and were known earlier. Examples: if $\lim s_n = L$, then $\lim_{x\to 1^-} (1-x)\sum_{n=0}^\infty s_n x^n = L$. The name comes from Niels Henrik Abel.
6.2 Comparison with Mercerian theorems
Mercerian theorems are Tauberian theorems in which no extra condition is needed: the summability method itself forces convergence. For example, the Cesàro method of order $-1$ (inversion of Cesàro) is Mercerian. These are rarer and usually rely on the method being “more powerful” than ordinary convergence in a specific sense.
6.3 Open problems and current research directions
Modern research in Tauberian theory includes:
- Optimal conditions: Determining the weakest possible Tauberian conditions for given methods.
- Multidimensional Tauberian theorems: For sequences indexed by $\mathbb{N}^d$ and integrals on $\mathbb{R}^d$.
- Nonlinear Tauberian theorems: Where the limit is not linear (e.g., for means of functions).
- Applications to spectral theory: Relationships between the asymptotic distribution of eigenvalues and the trace of the heat kernel, using Tauberian arguments (the Hardy–Littlewood–Karamata type).
- Connections with data analysis: Tauberian conditions in the context of summation of slowly varying sequences in machine learning and signal processing.