The Cesàro mean of a sequence $(s_n)$ is the arithmetic average $\sigma_N = \frac{1}{N+1}\sum_{n=0}^N s_n$. This is a **summability method**: a systematic way to assign a value to a sequence (or [series](/page/Series)) that may not converge in the ordinary sense. Cesàro summability is strictly weaker than ordinary convergence — every convergent sequence is Cesàro-convergent to the same limit, but some divergent [sequences](/page/Sequence) are also Cesàro-convergent. In Fourier analysis, the Cesàro means of the Fourier partial sums are the key to unlocking [Fejér's theorem](/theorems/584) and the [completeness of the trigonometric system](/theorems/585), bypassing the convergence difficulties of the [Dirichlet kernel](/page/Dirichlet%20Kernel). [motivation] ## Motivation ### Why Average? Many sequences of interest oscillate without converging, yet their oscillation has a well-defined "average behaviour." The simplest example is $s_n = (-1)^n$: the partial sums of Grandi's series $1 - 1 + 1 - 1 + \cdots$ alternate between $1$ and $0$, but their averages $\sigma_N = (N+1)^{-1}\sum_{n=0}^N s_n$ converge to $1/2$. More importantly, the Fourier partial sums $S_N f(x)$ of a [continuous](/page/Continuity) [function](/page/Function) may oscillate at certain points (due to the oscillating [Dirichlet kernel](/page/Dirichlet%20Kernel)), but their Cesàro means $\sigma_N f(x) = (N+1)^{-1}\sum_{n=0}^N S_n f(x)$ always converge uniformly — because averaging replaces the Dirichlet kernel with the positive [Fejér kernel](/page/Fejér%20Kernel). ### Regularity: Averaging Does Not Destroy Convergence The essential property of Cesàro summability is **regularity**: if the original sequence already converges, the Cesàro means converge to the same limit. This means Cesàro summability is a *strengthening* of ordinary convergence — it captures everything ordinary convergence captures, and more. Without regularity, using Cesàro means in Fourier analysis would be pointless: we would not know that the Cesàro means recover $f$ at points where the [Fourier series](/page/Fourier%20Series) already converges. ### Scope of This Page This page develops Cesàro means as a general tool in analysis: the definition, the regularity theorem, examples of Cesàro summability for divergent sequences, and the connection to Fourier analysis. The specific Fourier application — the [Fejér kernel](/page/Fejér%20Kernel) and Fejér's theorem — is developed on the dedicated kernel page. [/motivation] ## Definition The construction is elementary: replace a sequence by the running average of its first $N+1$ terms. [definition: Cesàro Mean] Let $(s_n)_{n \geq 0}$ be a sequence of real (or complex) numbers. The **Cesàro mean** of order $N$ is \begin{align*} \sigma_N := \frac{1}{N+1}\sum_{n=0}^N s_n. \end{align*} The sequence $(s_n)$ is **Cesàro summable** (or **(C,1)-summable**) to a limit $L$ if $\sigma_N \to L$ as $N \to \infty$. [/definition] The notation $(C,1)$ refers to the first-order Cesàro method; higher-order Cesàro means $(C, k)$ are obtained by iterating the averaging process $k$ times. We treat only the $(C, 1)$ case here. When $(s_n) = (S_n f(x))$ are the Fourier partial sums of a function $f$ at a point $x$, the Cesàro mean $\sigma_N$ is the **Fejér sum** $\sigma_N f(x) = \frac{1}{N+1}\sum_{n=0}^N S_n f(x)$, which can be written as the [convolution](/page/Convolution) $\sigma_N f = f * F_N$ with the [Fejér kernel](/page/Fejér%20Kernel). ## Regularity The most important theoretical property of Cesàro summability is that it is a *regular* summability method: it preserves [limits](/page/Limit). [quotetheorem:590] The proof splits $\sigma_N - L = \frac{1}{N+1}\sum(s_n - L)$ into a "head" (finitely many early terms that are bounded but diluted by $1/(N+1)$) and a "tail" (later terms that are each close to $L$). The head vanishes because any fixed sum divided by $N+1$ tends to zero; the tail is small because it averages quantities close to zero. The converse fails: a sequence can be Cesàro summable without converging. The averaging "smooths out" oscillations that prevent ordinary convergence. The following examples illustrate both directions. [example: Grandi's Series] Consider the series $1 - 1 + 1 - 1 + \cdots$ with partial sums $s_n = \sum_{k=0}^n (-1)^k$. For even $n$: $s_n = 1$; for odd $n$: $s_n = 0$. The sequence $(s_n) = 1, 0, 1, 0, 1, 0, \ldots$ does not converge. The Cesàro means are: \begin{align*} \sigma_N = \frac{1}{N+1}\sum_{n=0}^N s_n = \frac{\lceil (N+1)/2 \rceil}{N+1} \to \frac{1}{2} \quad \text{as } N \to \infty, \end{align*} since among $s_0, \ldots, s_N$ roughly half the terms equal $1$ and half equal $0$. The series $\sum (-1)^n$ is Cesàro summable to $1/2$, which agrees with the Abel sum $\sum (-1)^n x^n = 1/(1+x) \to 1/2$ as $x \to 1^-$. [/example] [example: Regularity In Action] Let $s_n = 1 + (-1)^n/n$ for $n \geq 1$, with $s_0 = 2$. Then $s_n \to 1$ (ordinary convergence). The Cesàro means are: \begin{align*} \sigma_N = \frac{1}{N+1}\sum_{n=0}^N s_n = 1 + \frac{1}{N+1}\sum_{n=0}^N \frac{(-1)^n}{n+1} + O(1/N). \end{align*} The alternating sum $\sum (-1)^n/(n+1)$ has bounded partial sums (it converges to $\log 2$), so its average is $O(1/N)$. Therefore $\sigma_N \to 1 = \lim s_n$, confirming regularity. [/example] [example: A Divergent Sequence That Is Not Cesàro Summable] Let $s_n = n$. Then $\sigma_N = \frac{1}{N+1}\sum_{n=0}^N n = \frac{N}{2} \to \infty$. Cesàro summability does not rescue sequences that diverge to infinity — it only handles oscillatory divergence. More precisely, if $s_n \to +\infty$ then $\sigma_N \to +\infty$ as well (this is another consequence of regularity, extended to the case $L = +\infty$). [/example] ## Connection to Fourier Analysis The power of Cesàro summability in Fourier analysis comes from the fact that the Cesàro mean of Fourier partial sums is a convolution with a *positive* kernel. The Fourier partial sums are $S_n f(x) = (f * D_n)(x)$, where $D_n$ is the [Dirichlet kernel](/page/Dirichlet%20Kernel). The Cesàro mean is: \begin{align*} \sigma_N f(x) = \frac{1}{N+1}\sum_{n=0}^N S_n f(x) = \frac{1}{N+1}\sum_{n=0}^N (f * D_n)(x) = f * \left(\frac{1}{N+1}\sum_{n=0}^N D_n\right)(x) = (f * F_N)(x), \end{align*} where $F_N = \frac{1}{N+1}\sum_{n=0}^N D_n$ is the [Fejér kernel](/page/Fejér%20Kernel). The exchange of convolution and finite averaging uses the linearity of convolution. The Fejér kernel is non-negative (its closed form is a ratio of $\sin^2$ terms), so the convolution $f * F_N$ is a genuine weighted average of $f$ — it cannot overshoot or undershoot the range of $f$. [Fejér's Theorem](/theorems/584) then asserts that $\sigma_N f \to f$ uniformly for $f \in C(\mathbb{T})$, which implies the density of trigonometric polynomials in $C(\mathbb{T})$ and ultimately the [completeness of the trigonometric system](/theorems/585) in $L^2(\mathbb{T})$. The logical chain is: **Cesàro averaging** (this page) → **positivity of the Fejér kernel** ([Fejér Kernel](/page/Fejér%20Kernel)) → **Fejér's theorem** → **completeness** → **$L^2$ convergence and Parseval** ([Fourier Series (Trigonometric)](/page/Fourier%20Series%20(Trigonometric))). ## Cesàro Means of Series The Cesàro mean is often applied not to a sequence $(s_n)$ directly but to the partial sums $s_n = \sum_{k=0}^n a_k$ of a series $\sum a_k$. The Cesàro mean of the partial sums can be rewritten in terms of the original series terms using **Abel summation** (summation by parts): \begin{align*} \sigma_N = \frac{1}{N+1}\sum_{n=0}^N s_n = \frac{1}{N+1}\sum_{n=0}^N \sum_{k=0}^n a_k = \sum_{k=0}^N a_k \cdot \frac{N + 1 - k}{N + 1} = \sum_{k=0}^N \left(1 - \frac{k}{N+1}\right) a_k. \end{align*} The weight $1 - k/(N+1)$ is a **triangular window**: it equals $1$ at $k = 0$ and decreases linearly to $0$ at $k = N + 1$. This is exactly the Fourier coefficient of the [Fejér kernel](/page/Fejér%20Kernel): $\hat{F}_N(k) = 1 - |k|/(N+1)$. The triangular window attenuates the later (high-frequency) terms, smoothing the partial sums and suppressing the oscillations that cause convergence difficulties. ## References - Hardy, G. H. (1949). *Divergent Series*. Oxford University Press. - Katznelson, Y. (2004). *An Introduction to Harmonic Analysis* (3rd ed.). Cambridge University Press. - Stein, E. M. and Shakarchi, R. (2003). *Fourier Analysis: An Introduction*. Princeton University Press.