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Fourier Analysis
Definitions
Fourier Series
- Fourier coefficients: \begin{align*}\widehat{f}(n)=a_{n}=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(\theta) e^{-i n \theta} d \theta, \quad n \in \mathbb{Z}\end{align*}
- Partial sums: \begin{align*}S_N f(\theta) \sim \sum_{{\left\lvert {n} \right\rvert} \leq N} a_{n} e^{i n \theta}\end{align*}
- Fourier series: \begin{align*}f(\theta) \sim \sum_{n=-\infty}^{\infty} a_{n} e^{i n \theta} = \lim_{N\to \infty} S_N f(\theta), \qquad c_n = \widehat{f}(n)\end{align*}
- Parseval: \begin{align*}|f|_{L^{2}(-\pi, \pi)}^{2}=\int_{-\pi}^{\pi}|f(x)|^{2} d x=2 \pi \sum_{n=-\infty}^{\infty}\left|c_{n}\right|^{2}\end{align*}
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Compuattions of Fourier expansions:
- \begin{align*}f(\theta) \coloneqq{1\over 4}(\pi - \theta)^2 \implies f(\theta) \sim \frac{\pi^{2}}{12}+\sum_{n=1}^{\infty} \frac{\cos n \theta}{n^{2}}\end{align*}
- \begin{align*} f(\theta)=\frac{\pi}{\sin \pi \alpha} e^{i(\pi-\theta) \alpha} \implies f(\theta) \sim \sum_{n=-\infty}^{\infty} \frac{e^{i n \theta}}{n+\alpha} \end{align*}
- Applying Parseval: \begin{align*}f(x) = x \implies c_n ={(-1)^{n}}{}\cdot {i\over n} \implies \zeta(2) \coloneqq\sum_{n=1}^{\infty} \frac{1}{n^{2}}=\frac{\pi^{2}}{6}\end{align*}
Convolutions
- Convolution: \begin{align*}(f * g)(x)=\frac{1}{2 \pi} \int_{-\pi}^{\pi} f(y) g(x-y) d y\end{align*}
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Dirichlet kernel:
\begin{align*}D_{N}(x)=\sum_{n=-N}^{N} e^{i n x}=\frac{\sin \left(\left(N+\frac{1}{2}\right) x\right)}{\sin (x / 2)}\end{align*}
- Fourier series as a convolution: \begin{align*}S_N f(x) =\left(f * D_{N}\right)(x)\end{align*}
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Poisson kernel:
\begin{align*}P_{r}(\theta)=\sum_{n=-\infty}^{\infty} r^{|n|} e^{i n \theta} = \frac{1-r^{2}}{1-2 r \cos \theta+r^{2}}\end{align*}
- Arises in solution of steady-state temperature of heat on a plate.
Convergence
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Big ${ \mathsf{O}} $ notation:
\begin{align*}f\in { \mathsf{O}} (g) \iff \exists x_0 \in X, M\in {\mathbf{R}}, x\geq x_0 \implies {\left\lvert {f(x)} \right\rvert} \leq M{\left\lvert {g(x)} \right\rvert}\end{align*}
- \(f\in { \mathsf{O}} (1) \implies f\) is bounded.
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Classes of functions:
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\(C^0\): continuous functions
- Holder continuous: \begin{align*}|f(x)-f(y)| \leq C|x-y|^{\alpha}\end{align*}
- A general Holder condition: \begin{align*}\sup _{\theta}|f(\theta+t)-f(\theta)| \leq A|t|^{\alpha} \quad \text { for all } t\end{align*}
- Lipschitz cts: \begin{align*}d_{Y}\left(f\left(x_{1}\right), f\left(x_{2}\right)\right) \leq C\cdot d_{X}\left(x_{1}, x_{2}\right)\end{align*}
- \(C^n\): \(n\) times differentiable with \(f^{(n)}\) continuous.
- Piecwise linear functions
- Almost everywhere variants of all of the above.
- Riemann integrable: \(f\) is bounded and there exists a partition with \({\left\lvert {U - L } \right\rvert} < {\varepsilon}\)
- Lebesgue integrable
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\(C^0\): continuous functions
- \(L^p\) norms: \begin{align*}{\left\lVert {f} \right\rVert}_{L^p} \coloneqq\qty{\int_X {\left\lvert {f} \right\rvert}^p \,d\mu}^{1\over p}, \qquad {\left\lVert {f} \right\rVert}_{L^\infty} = {\mathrm{ess}}\sup{f} = \inf \{C \geq 0:|f(x)| \leq C \text { for almost every } x\}\end{align*}
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Types of convergence:
- Pointwise: \begin{align*}{\left\lvert {f_n(x) - F(x) } \right\rvert} \overset{n\to\infty}\longrightarrow 0.\end{align*}
- Pointwise a.e.: \begin{align*}M_n \coloneqq\mu\left\{{x\in X {~\mathrel{\Big\vert}~}f_n(x) \not\to x}\right\} \overset{n\to\infty}\longrightarrow 0\end{align*}
- Uniform: \begin{align*}\sup \left\{{{\left\lvert {f_n(x) - f(x)} \right\rvert} {~\mathrel{\Big\vert}~}x\in X}\right\} \overset{n\to\infty}\longrightarrow 0\end{align*}
- In sup norm \(L^\infty\): \begin{align*}{\left\lVert {f_n - f} \right\rVert}_{L^\infty} \overset{n\to\infty}\longrightarrow 0\end{align*}
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\(L^2\):
\begin{align*}{\left\lVert {f_n - f} \right\rVert}_{L^2} \overset{n\to\infty}\longrightarrow 0\end{align*}
- Explicit \(L^2\), i.e. mean square: \begin{align*}\frac{1}{2 \pi} \int_{-\pi}^{\pi}\left|S_{N}(f)(\theta)-f(\theta)\right|^{2} d \theta \overset{N\to\infty}\longrightarrow 0\end{align*}
- Convergence in measure: \begin{align*}\lim _{n \rightarrow \infty} \mu\left(\left\{x \in X:\left|f(x)-f_{n}(x)\right| \geq \varepsilon\right\}\right)\overset{n\to\infty}\longrightarrow 0\end{align*}
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Theorems:
- The uniform limit of continuous functions is continuous.
- \(f\) is Riemann integrable iff \(f\) is bounded and has null discontinuity set. Necessary because \(f(x) = \chi_{\mathbf{Q}}(x)\) is not integral on \([0, 1]\) since it is discts at every irrational.
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If \(f\) is cts and \(\left\{{K_n}\right\}\) is a good kernel,
\begin{align*}(f\ast K_n) \overset{n\to\infty}\longrightarrow f\end{align*}
- The Dirichlet kernel is not a good kernel!
- If \(f_n\to f\) uniformly on a compact set \(A\) then \(\int_A f_n\to \int_A f\). Compactness is necessary: take \(f_n(x) \coloneqq\chi_{[0, n]}(x)\cdot {1\over n}\).
- If \(f: S^1\to {\mathbf{R}}\) is integrable and \(\widehat{f}(n) = 0\) for all \(n\), then \(f(x)= 0\) on the set of continuity of \(f\).
- If \(f:S^1\to {\mathbf{R}}\) is cts and \(\left\{{ \widehat{f}(n)}\right\} \in \ell^1({\mathbf{Z}})\) (so the Fourier series converges absolutely), then \(\widehat{f} \to f\) uniformly.
- If \(f\in C^2(S^1)\), then \(\widehat{f}(n) \in { \mathsf{O}} ({\left\lvert {n} \right\rvert}^{-2})\) and \(\widehat{f}\) converges absolutely and \(S_N f \to f\) uniformly.
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Riemann-Lebesgue
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Fourer inversion:
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Properties of the Fourier transform:
Common counterexamples
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A discontinuous Riemann-integrable function: \(\chi_{[0, 1]} - \chi_{\left\{{1\over 2}\right\}}\)
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A Riemann-integrable function with countably infinitely many discontinuities:
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Convergence comparisons:
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Exercises
- \(\mathcal{F}\left(f^{\prime}\right)(\xi)=2 \pi i \xi \cdot \mathcal{F}(f)(\xi)\)
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\(\widehat{f\ast g}(n) = \widehat{f}(n) \cdot \widehat{g}(n)\):
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Show that the partial sums of Fourier coefficients are given by convolution against the Dirichlet kernel:
