Subjects: Fourier Analysis
Links: Hilbert Spaces, Good Kernels and Convergence in Fourier Analysis, Lp spaces

In $L^2([0,2\pi ])$

Let's consider the $L^2([0,2\pi ])$, then we know that the inner product is of the form $$\langle f, g\rangle = \frac{1}{2\pi} \int_0^{2\pi} f(\theta) \overline{g(\theta)} , d\theta$$and the norm $$|f|^2 = \frac{1}{2\pi} \int_0^{2\pi}|f(\theta)|^2, d\theta$$
For each integer $n$, let $e_n(\theta )=e^{in\theta }$, and observe that that the family $\{e_n{\}}_{n\in \mathbb{Z}}$ is orthonormal; that is $$\langle e_n, e_m\rangle = \delta_{n,m}$$with ${\delta }_{n,m}$ being the Kronecker delta.

Let $f$ be integrable function on the circle and let $a_n$ denote its Fourier coefficients. We see that the Fourier coefficients are represented by inner products of $f$ with the elements in the orthonormal set $\{e_n{\}}_{n\in \mathbb{Z}}$: $$\langle f, e_n\rangle = \frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-in\theta}\ d\theta = a_n$$
We see that $S_N(f)=\sum _{|n|\le N}{a}_n{e}_n$. Then the orthonormal property of the family $\{e_n\}$ and the fact that $⟨f,e_n⟩$ imply that $f-\sum _{|n|\le N}{a}_n{e}_n$ is orthogonal to $e_n$, for all $|n|\le N$. There for we have that $$(f- S_N(f)) \perp \sum_{|n| \le N} b_n e_n$$for any complex numbers $b_n$. From this fact we get that, $f=(f-S_N(f))+S_N(f)$, and $$|f|^2 = |f-S_N(f)|^2+\left|\sum_{|n| \le N}a_n e_n\right|^2$$by the orthonormality we have that $$\left|\sum_{|n| \le N}a_n e_n\right|^2 = \sum_{|n| \le N}|a_n|^2$$, we deduce that $$|f|^2 = |f-S_N(f)|^2+\sum_{|n| \le N}|a_n|^2$$
Best Approximation: If $f$ is integrable on the circle with Fourier coefficients $a_n$, then $$|f-S_N(f)| \le \left|f-\sum_{|n|\le N} c_n e_n\right|$$for any complex numbers $c_n$. Moreover, equality holds precisely when $c_n=a_n$ for all $|n|\le N$.

Th: Let $f$ be an integrable function on the circle with $f\sim \sum _{n\in \mathbb{Z}}{a}_n{e}^{in\theta }$. Then we have that:

• Mean-square convergence of the Fourier series: $$\frac{1}{2\pi} \int_0^{2\pi} |f(\theta)- S_N(f)(\theta)|^2 , \theta \to 0 \qquad N\to \infty$$or $S_N(f)\to f$ in $L^2([0,2\pi ])$
• Parseval's identity $$\sum_{n = -\infty}^\infty |a_n|^2 = \frac{1}{2\pi} \int_0^{2\pi}|f(\theta)|^2, d\theta$$In general, we can make $f$ be $L$ periodic then get $$\sum_{n = -\infty}^\infty |a_n|^2 = \frac{1}{L} \int_0^{L}|f(\theta)|^2, d\theta$$
  Bessel's Inequality: Let $\{e_n\}$ is any orthogonal family of function on the circle, and $a_n=⟨f,e_n⟩$, then we may deduce that $$\sum_{n = -\infty}^\infty |a_n|^2 \le |f|^2$$
  We have associated every element of $L^2([0,2\pi ])$ with a sequence $\{a_n\}$, and Parseval's identity guarantees that $\{a_n\}\in {\ell }^2(\mathbb{Z})$.

Riemann-Lebesgue Lemma If $f$ is integrable on the circle, then $\hat{f}(n)\to 0$ as $|n|\to \mathrm{\infty }$. An equivalent reformulation, if that if $f$ is integrable in $[0,2\pi ]$, then $$\int_0^{2\pi} f(\theta) \sin(N\theta), d\theta \to 0 \qquad N \to \infty$$and $$\int_0^{2\pi} f(\theta) \cos(N\theta), d\theta \to 0 \qquad N \to \infty$$
Cor: Let $f$ be a periodic and of class ${\mathcal{C}}^k$, then $$\hat f(n) = o(1/|n|^k)$$

Lemma: suppose that $F$ and $G$ are integrable on the circle with $$F \sim \sum a_m e^{in\theta} \quad \text{and}\quad G \sim \sum b_n e^{in\theta}$$
Then $$\frac{1}{2\pi} \int_0^{2\pi} F(\theta) \overline{ G(\theta)}, d\theta = \sum_{n = -\infty}^\infty a_n \overline{b_n}$$
and if we a make $F$ and $G$ be $L$ periodic, we get that $$\frac{1}{L} \int_0^L F(\theta) \overline{ G(\theta)}, d\theta = \sum_{n = -\infty}^\infty a_n \overline{b_n}$$

With this we have more things to prove the absolute convergence and hence uniform convergence.

We say that $f$ satisfies a Hölder condition of order $\alpha$, namely $$|f(x)-f(y)| \le C|x-y|^\alpha $$for some $0<\alpha \le 1$, and some $C>0$ and all $x,y$. Then $$\hat f(n) = O(1/|n|^\alpha)$$We note that for $\alpha =1$, then $f$ is Lipschitz continuous.

Bernstein Theorem: If $f$ satisfies a Hölder condition of order $\alpha >1/2$, then the Fourier series of $f$ converges absolutely.

Pointwise Convergence

Th: Let $f$ be an integrable function on the circle which is differentiable at a point ${\theta }_0$. Then $S_N(f)({\theta }_0)\to f({\theta }_0)$ as $N\to \mathrm{\infty }$.

We can weaken further if we only assume that $f$ satisfies a Lipschitz condition at ${\theta }_0$; that is $$|f(\theta) - f(\theta_0) \le |\theta - \theta_0|$$For some $M\ge 0$ and all $\theta$.

Riemann Localisation Theorem: Suppose $f$ and $g$ are two integrable functions defined on the circle, and for some ${\theta }_0$ there exists an open interval containing ${\theta }_0$ such that $$f|_I = g|_I$$then $S_N(f)({\theta }_0)-S_N(g)({\theta }_0)\to 0$ as $N\to \mathrm{\infty }$.

We also have that near a discontinuity, we have that the Fourier series tend to overshoot/undershoot, by around 9% of the jump at this is called Gibb's phenomenon