\(L^2\) convergence of Fourier Series

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Let us approximate a function \(f(x)\) by a linear combination of \(\{e^{ikx} | k = -n, \cdots, 0, \cdots, n\}\). We show that the partial Fourier sum is the ``best'' of such approximators. Then, we prove that the Fourier series of the function \(f\) converges to \(f\) in the \(L^2\) norm.



Lemma (``best'' approximator)

Let any \(n\in\mathbb{N}\), consider the set of \(2n + 1\) functions of the form
\[\varphi_k(x) = e^{ikx} ~~~ (|k| \leq n).\]
Let \(v_n\) be an arbitrary linear combination of these functions:
\[v_n(x) = \sum_{|k|\leq n}\gamma_k \varphi_k(x) \]
where \(\gamma_k \in\mathbb{C}\) are constants. Suppose \(f\in C_{2\pi}^{0}\). Then,
\[\|f - S_n\| \leq \|f - v_n\|\]
where \(S_n = S_n[f]\) is the partial sum of the Fourier series \(S[f]\). In other words, \(S_n[f]\) is the best approximator of \(f\) among all the linear combinations of \(\{\varphi_{k}\}_{|k|\leq n}\) with respect to the \(L^2\) norm.
Proof. 
\[\begin{eqnarray*} \|f - v_n\|^2 & = & \|(f - S_n) + (S_n - v_n)\|^2\\ & = & \|f - S_n\|^2 + 2\Re(f-S_n, S_n- v_n) + \|S_n - v_n\|^2. \end{eqnarray*}\]
But the inner product \((f-S_n, S_n- v_n) = 0\) because \(S_n - v_n\) is a linear combination of \(\varphi_k\) and
\[\begin{eqnarray*} (f - S_n, \varphi_k) &=& (f, \varphi_k) - (S_n, \varphi_k)\\ &=& 2\pi c_k - 2\pi c_k = 0. \end{eqnarray*}\]
Therefore,
\[\|f - v_n\|^2 = \|f - S_n\|^2 + \|S_n - v_n\|^2 \geq \|f - S_n\|^2\]
where the equality holds if and only if \(v_n = S_n\). ■

Definition (\(L^2\) convergence)

For the sequence of functions \(\{f_n\}\) and some function \(f_0\), \(\{f_n\}\) is said to converges to \(f_0\) in the \(L^2\) norm if
\[\|f_n - f_0\| \to 0 ~~~ (n \to \infty).\]

Example. Consider the functions \(u_n(x) = \sin(\pi x^n), n = 1, 2, \cdots\) on the interval \((0, 1)\). For each \(x\in (0,1)\), \(\lim_{n\to\infty}u_n(x) = 0\). But \(\{u_n\}\) does not converge uniformly to \(u_0(x) \equiv 0\). In fact, for \(x_n = \left(\frac{1}{2}\right)^{\frac{1}{n}}\), \(u_n(x_n) = 1\). On the other hand, with the change of variables \(t = x^n\),
\[\begin{eqnarray*} \|u_n\|^2 &=& \int_0^1|u_n(x)|^2dx = \int_0^1\sin^2(\pi x^n)dx = \int_0^1\sin^2(\pi t)\cdot\frac{t^{\frac{1}{n}-1}}{n}dt\\ &\leq& \frac{1}{n}\int_0^1(\pi t)^2t^{\frac{1}{n}-1} = \frac{\pi^2}{n}\int_0^1t^{\frac{1}{n}+1}dt = \frac{\pi^2}{n}\frac{1}{\frac{1}{n} + 2}\\ &=& \frac{\pi^2}{2n + 1} \to 0. \end{eqnarray*}\]
Thus, \(u_n(x)\) converges to \(u_{0}(x) \equiv 0\) in \(L^2\) (on the interval \((0,1)\)). □
Remark. We can define a distance function based on a norm. That is, the distance \(d(f,g)\) between two functions \(f\) and \(g\) can be defined by
\[d(f,g) = \|f - g\|.\]
Thus, the convergence in the \(L^2\) norm is a generalization of the convergence of a point sequence in \(\mathbb{R}^n\) (or \(\mathbb{C}^n\)).

Theorem (\(S[f]\) converges to \(f\) in the \(L^2\) norm)

If \(f\in C_{2\pi}^{0}\), then the Fourier series \(S[f]\) converges to \(f\) in the \(L^2\) norm.
Proof. For the partial sum \(S_n\) and Fejér partial sum \(\sigma_n\) of the Fourier series of \(f\) (see Kernels arising from the Fourier series), by the above Lemma (``best'' approximator) , we have
\[\|f - S_n\| \leq \|f - \sigma_n\|.\]
Note that \(\sigma_n\) is also a linear combination of \(\{e^{ikx}\}\). Furthermore, \(\sigma_n\) uniformly converges to \(f\) (Lemma (Uniform convergence of \(\sigma_n[f]\))) so that \(\|f - \sigma_n\| \to 0\). Hence \(\|f - S_n\| \to 0\). ■

This theorem can be further extended to include discontinuous functions. We list the theorem without proof.

Theorem

If \(f\in\mathcal{R}_{2\pi}^2\), then \(S[f]\) converges to \(f\) in the \(L^2\) norm.



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