---- > [!definition] Definition. ([[Abel summable]]) > The **Abel means** of the [[series]] $\sum_{k=0}^{\infty} c_{k}$ are $A(r) = \sum_{k=0}^{\infty} r ^{k} c_{k}.$ > We call the [[series]] $\sum_{k=0}^{\infty} c_{k}$ **Abel summable** if $\lim_{ r ↑ 1 } A(r)$ [[converge]]s. \ The **Abel means** of the [[Fourier series]] of $f$ is $A_{r}(f)(\theta)=\sum_{n \in \zz}^{} \hat{f}(n)e ^{in \theta} r ^{|n|}=(f * P_{r})(\theta)$(converges $\fa r$ by [[Weierstrass M-test]] [^1]), where $P_{r}$ is the [[good kernel|(good)]] [[Poisson kernel]]. (we showed this in IBL). By [[limit of good kernels approximates the convolution identity, given continuity]] we in turn have $\lim_{ r \uparrow 1 } A_r(f)(\theta_{0})=f(\theta_{0})$ provided $f$ is [[continuous]] at $\theta_{0}$; if $f$ is [[continuous]] then the [[converge|convergence]] is [[uniform convergence|uniform]] in $\theta_{0}$. [^1]: Because $|\hat{f}(n)|\leq\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(\theta)e ^{in \theta}|\,d\theta=\frac{1}{2\pi}\int_{-\pi}^{\pi}|f(\theta)| \, d\theta$ is [[uniformly bounded]] (RHS doesn't depend on $n$), in turn $\hat{f}(n)e ^{in \theta}$ is uniformly bounded, so $A_{r}f(\theta)$ is [[uniformly bounded]] by constant time a [[geometric series]]... ---- #### ---- #### 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 > ```