---- > [!definition] Definition. ([[good kernel]]) A family of [[Riemann integral|Riemann integrable]] functions $\{ k_{n} \}_{n=1}^\infty$ [[function on the (unit) circle|on the circle]] is called a **family of good kernels** if >1. $\frac{1}{2\pi} \int_{-\pi}^{\pi} k_{n}(x) \,dx=1 \ \ \fa n \in \zz$ (Unit mean) >2. $\ex M > 0$ s.t. $\|k_{n}\|_{1}=\frac{1}{2\pi}\int_{-\pi}^{\pi} |k_{n}(x)|\,dx \leq M$, $\fa n \in \zz$ ($\mathcal{L}^{1}$ [[seminorm|(semi)]][[Lp-norm|norm]] is uniformly bounded); >3. $\fa \delta>0$, $\lim_{ n \to \infty } \int_{\delta \leq |x| \leq \pi}^{} |k|\,dx=0$. (Concentrates near origin) ![[CleanShot 2023-04-07 at [email protected]]] > Note $(1) \implies (2)$ when $k_{n}(x) \geq 0$ $\fa x, \ \fa n$. > [!basicexample] > - [[Fejer Kernel]]s are **good kernels** (see 396 hw) > - [[Poisson kernel]]s are **good kernels** (see 396 hw) > [!basicnonexample] > - [[Dirichlet Kernel]]s are *not* **good kernels**. ---- #### ---- #### 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 > ```