[[Noteworthy Uses]]:: *[[Noteworthy Uses]]* [[Proved By]]:: [[limit of good kernels approximates the convolution identity, given continuity]], [[on the circle, continuous functions are dense in Riemann integrable functions with respect to the 1-seminorm]], [[Fourier series is Cesaro summable at points of continuity|trigonometric polynomials are dense (uniformly) in the continuous functions on the circle]] ---- - Denote by $\mathscr{R}$ the [[vector space]] of [[Riemann integral|Riemann integrable]] functions [[function on the (unit) circle|on the circle]]; - Let $f,g \in \mathscr{R}$ be arbitrary; - Denote by $S_{N}$ the $N^{th}$ [[Fourier series|Fourier partial sum]] of $f$, $S_{N}(\theta)=\sum_{n=-N}^{N}\hat{f}(n)e ^{in \theta}$[[Lp-norm|.]] > [!theorem] Theorem. ([[L2 convergence of Fourier Series]]) > We have > 1. $\lim_{ N \to \infty } \int _{-\pi}^{\pi} |f(\theta)-S_{N}(f)(\theta)|^{2} \, d\theta=0$, and thus $\{ e ^{in \theta} \}_{n \in \zz}^{}$ form an [[orthonormal basis]] for $\mathscr{R}$ ; > 2. (Parseval) $\frac{1}{2\pi}\int _{-\pi}^{\pi}|f(\theta)|^{2} \, d\theta = \sum_{n=-\infty}^{\infty}|\hat{f}(n)|^{2}$; > 3. (Plancherel) $\frac{1}{2\pi}\int_{-\pi}^{\pi} f(\theta)\overline{g(\theta)}\,d \theta=\sum_{n=-\infty}^{\infty} \hat{f}(n)\overline{\hat{g}(n)}$. > [!proof]- Proof. ([[L2 convergence of Fourier Series]]) > **1.** $f$ is [[Riemann integral|Riemann integrable]] and hence bounded by $M>0$. Fix $\varepsilon >0$. > The idea is to approximate $f$ in two steps: we first [[on the circle, continuous functions are dense in Riemann integrable functions with respect to the 1-seminorm|approximate it in the L1 sense with a continuous function on the circle]], and then [[Fourier series is Cesaro summable at points of continuity|approximate the result in a uniform sense with a trigonometric polynomial on the circle]]. So... $\ex g$ [[continuous]] [[function on the (unit) circle|on the circle]] s.t. $|g(\theta)|\leq|f(\theta)| \leq M$ and $\int _{-\pi}^{\pi} |f(\theta)-g(\theta )| \, d\theta \leq \frac{\pi}{4M} \varepsilon^{2}$. Now we compute wrt the [[Lp-norm|L2 norm]] $\begin{align} > \|f-g\|:= & \left( \frac{1}{2\pi} \int _{-\pi} ^{\pi} |f(\theta)-g(\theta)|^{2} \, d\theta \right) ^{1/2} \\ > \leq & \left(\frac{2M}{2\pi} \int_{-\pi}^{\pi }|f(\theta)-g(\theta)|\, d \theta \right)^{1/2} \\ > \leq & \left( \frac{2M}{2\pi} \frac{\pi}{4M} \varepsilon^{2} \right) \\ > = & \frac{\varepsilon}{2}, > \end{align}$ > During our work with [[Fejer Kernel|Fejer kernels]] we obtained [[Fourier series is Cesaro summable at points of continuity|this result]] which we use now to obtain a [[trigonometric polynomial]] $g$ s.t. $|g(\theta)-p(\theta)| \leq \frac{\varepsilon}{2} \ \fa \theta$. Then we can bound the L2 distance $\|g-p\|$ as $\begin{align} > \|g-p\| = & \left( \frac{1}{2\pi} \int_{-\pi}^{\pi} |g(\theta)-p(\theta)|^{2}\, d\theta \right) ^{1/2} \\ > \leq & \left( \frac{1}{2\pi} \left( \frac{\varepsilon}{2} \right)^{2} 2\pi \right) \\ > \leq & \frac{\varepsilon}{2}. > \end{align}$ Now by the [[triangle inequality]] $\|f-p\| \leq \varepsilon$, where $p(\theta)=\sum_{n=-N_{0}}^{N_{0}}$ for some $N_{0} \in \nn$. Consider $N>N_{0}$. By definition $p$ is some element in [[submodule generated by a subset]]$\{ e ^{-iN \theta},\dots, e ^{iN \theta} \}$. Recall that $S_{N}(f)(\theta)=\sum_{n=-N}^{N} \langle f, e_{n} \rangle e_{n}$; now by [[minimum distance to a subspace is the orthogonal projection onto it]] we get $\|f-S_{N}(f)\|\leq \|f-p\|$ > which is less than $\varepsilon$ by the above argument. $\qedin$ > > **2**. Let $f \in \mathscr{R}$. Recall that $S_{N}(f)$ is the [[orthogonal projection]] of $f$ onto $\span(e_{-N},\dots,e_{N})$, and as such we have $f - S_{N}(f) \perp S_{N}(f)$. Then $\begin{align} > \|f\|^{2} = & \|f-S_{N}(f)+S_{N}(f)\|^{2} \\ > = & \|f-S_{N}(f)\|^{2}+\|S_{N}(f)\|^{2} \\ > = & \|f-S_{N}(f)\| ^{2} + \sum_{n=-N}^{N} |\hat{f}(n)|^{2}. > \end{align}$ > Now taking $\lim_{ N \to \infty }$ to both sides yields the result since the first term converges to $0$ by **1**. [^1]: ---- #### [^1]: In the second step here we wrote $|f(\theta)-g(\theta)|^{2}=|f(\theta)-g(\theta)| \ |f(\theta)-g(\theta)| \leq 2M |f(\theta)-g(\theta)|$, using that because $M$ bound $f$ and $g$, their difference is at most $2M$. ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as 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 > ```