---- > [!definition] Definition. ([[Cesaro summable]]) > Define the **$N ^{th}$ Cesaro sum** of the [[series]] $\sum_{k=0}^{\infty} c_{k}$ as $\sigma_{N} := \frac{s_{0} + s_{1} + \dots + s_{N-1}}{N}, \text{ where } s_{N} = \sum_{k=0}^{N} c_{n}.$ > We call the series $\sum_{k=0}^{\infty} c_{k}$ **Cesaro Summable** if $\lim_{ N \to \infty } c_{n}$ [[converge]]s. > \ > **Remark.** If the $S_{N}$ are [[Fourier series|Fourier partial sums]], then $\sigma_{N}(f)=f * F_{N}$— see below. \ **Remark 2.** If the $S_{N}$ are [[Fourier series|Fourier partial sums]] ($S_{N}=\sum_{k=-N}^{N} \hat{f}(k) e ^{ik \theta}$ ), then $\sigma_{N}(f)(\theta)=\frac{1}{N}\sum_{n=0}^{N-1}\sum_{k=-n}^{n} \hat{f}(k) e ^{ik \theta}$ the point is that $\sigma_{N}$ is a [[linear combination]] of $\hat{f}(k)$. > [!justification] > **Note.** Recall the definition of [[Dirichlet Kernel]]. Noting [^1] that $s_{N}=f * D_{N}$, Using [[linear map|linearity]] of [[convolution]] $\sigma_{N}(f)(\theta)=\frac{1}{N}\sum_{n=0}^{N-1}(f * D_{n})(\theta)=\big(f * (\frac{1}{N}\sum_{n=0}^{N-1}D_{n})\big)(\theta )$. Thus $\sigma_{N}(f)=f * F_{N}(f)$ where $F_{N}$ is the $N^{th}$ [[Fejer Kernel]]. [^1]: Let's check this. $(f * D_{N})(\theta)=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi)D_{N}(\theta-\varphi)\,d \varphi=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi)\sum_{n=-N}^{N}e^{in(\theta-\varphi)}\,d \varphi$ and pulling out the sum gives $\sum_{n=-N}^{N}\big(\frac{1}{2\pi}\int_{-\pi}^{\pi}f(\varphi) e ^{-in \varphi}\,d \varphi\big)e ^{in \theta}=\sum_{n=-N}^{N}\hat{f}(n)e ^{in \theta}=S_{N}(\theta)$. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```