# Mean-square convergence

Theorem 3.2.1 Suppose $f\in \mathcal R(S^1)$, then we have mean-square convergence of the Fourier series

$$\dfrac 1{2\pi}\int ^\pi_{-\pi}|f(\theta)-S_N(f)(\theta)|^2\mathrm d\theta \to 0,\text{ as }N\to \infty$$

Before the proof of the Theorem 3.2.1, we consider the space $\mathcal R$, then let

$$e_n(\theta)=e^{in\theta},\ \forall n\in\mathbb N$$

and obviously

$$(e_n,e_m)=\delta_{nm}$$

$$(f,e_n)=\dfrac1{2\pi}\int^\pi_{-\pi}f(\theta)e^{-in\theta}\mathrm d\theta=\hat f(n)$$

Now we have the best approximation of the Fourier series

Lemma 3.2.2 If $f\in \mathcal R(S^1)$, then

$$||f-S_N(f)||\leq ||f-\sum_{|n|\leq N}c_ne_n||$$

for any complex numbers $c_n$. Moreover, equality holds precisely when $c_n=\hat f(n)$ for all $|n|\leq N$

Theorem 3.2.3 If $f\in\mathcal R(S^1)$, then we have Parseval's identity

$$\sum^\infty_{n=-\infty}|\hat f(n)|^2=||f||^2=\dfrac{1}{2\pi}\int^\pi_{-\pi}|f(\theta)|^2\mathrm d\theta$$

that is to say

$$||f||^2=\dfrac14a_0^2+\dfrac12\sum^\infty_{n=1}a^2_n+b_n^2$$

This identity provides an important connection between the norms in the two vector spaces $l^2(\mathbb Z),\mathcal R$.

Corollary 3.2.4 If

Theorem 3.2.5 If $f\in\mathcal R(S^1)$, then we have Riemann-Lebesgue lemma

$$\hat f(n)\to 0,\text{ as }|n|\to\infty$$

an equivalent reformulation of this theorem is that

$$\int ^\pi_{-\pi}f(\theta)\sin (N\theta)\mathrm d\theta\to 0,\text{ as } N\to\infty$$

$$\int^\pi_{-\pi}f(\theta)\cos (N\theta)\mathrm d\theta\to 0,\text{ as } N\to\infty$$

At last, we give a more general version of the Parseval identity

Lemma 3.2.6 Suppose $F,G\in\mathcal R(S^1)$ with

$$F\sim \sum a_ne^{in\theta},\quad G\sim\sum b_ne^{in\theta}$$

then

$$\dfrac1{2\pi}\int^\pi_{-\pi} F(\theta)\overline{G(\theta)}\mathrm d\theta=\sum^\infty_{n=-\infty}a_n\overline{b_n}$$

Theorem 3.2.7 Suppose $f\in\mathcal R(S^1)$, with $f\sim \displaystyle \sum A_ne^{inx}$, then for all $(a,b)\subset [-\pi,\pi]$, then

$$\dfrac 1{2\pi}\int^b_af(x)\mathrm dx=\sum_{n\in\mathbb Z} \int^b_a A_ne^{inx}\mathrm dx$$

# Pointwise convergence