I'm learning about Fourier Series from Carother's Real Analysis. I'm ultimately trying to prove the following result:
Let f $\in$ $R([-\pi,\pi])$, then ||$f$ - $S_{n}(f)||_{2}$ $\overset{n \to \infty}{\longrightarrow} 0$
in order to justify the use of Fourier Series representation of a function. However, prior to showing this we must show the following:
Let $f$ $\in$ $R([-\pi,\pi$]), then $\exists g$ $\in C^{2\pi}$ $\ni$ $||f-g||_{2} < \epsilon$.
I have been trying to use Weiestrass's Second Approximation Theorem to find a common Trigometric polynomial, then I was thinking about using the triangle inequality. However, when I use the triangle inequality I end up with the distance between both approximating polynomials for $f$ and $g$. For example,
Let $f \in C^{2\pi}$, the set of all continuous $2\pi$-Periodic functions on $[-\pi,\pi]$, by Weiestrass's Second Approximation Theorem there exists $T_{1} \in T_{n}$ such that $||f-T_{1}||_{\infty} < \epsilon$. Similarly, there exists $T_{2} \in T_{m}$ such that $||g-T_{2}||_{\infty} < \epsilon$. Therefore, we have the following: \begin{align} ||f-g||_{\infty} &\leq ||f-T_{1}||_{\infty} + ||T_{1} - g|| \\ &\leq ||f-T_{1}||_{\infty} + ||T_{1} - T_{2}||_{\infty} + ||T_{2} - g||_{\infty} \end{align} I know that $||f-T_{1}||_{\infty} < \epsilon$ and $||T_{2} - g||_{\infty} < \epsilon$, however I don't know how to deal with the middle term $||T_{1} - T_{2}||_{\infty}$.
I was then going to use the fact that the $L_{2}$-norm is bounded above by the sup-norm. Specifically
\begin{align} ||f-g||_{2} \leq \sqrt{2}||f-g||_{\infty} \end{align}
Hence, $||f-g||_{\infty} < \frac{\epsilon}{\sqrt{2}} \implies ||f-g||_{2} < \epsilon$ to complete the proof.
If my approach is incorrect please let me know. Any help is greatly appreciated. Thanks.
Remark: You don't need to use very powerful machinery to prove this statement, i.e. Weierstrass's second approximation theorem.
Hint: Fix $\varepsilon>0$. Since $f$ is riemann integrable, then there exists a partition of $[-\pi, \pi]$ say $\mathcal{P}=\{x_0=-\pi, x_1, \ldots, x_{N-1}, x_N=\pi\}$ such that $\sum^N_{i=1} M_i\Delta x_i-\sum^N_{i=1} m_i\Delta x_i<\varepsilon$ where $M_i$ is the supremum on $[x_{i-1}, x_i]$ and $m_i$ the infimum. Then define \begin{align} m_i\leq g(t) := \frac{x_i-t}{\Delta x_i}f(x_{i-1})+\frac{t-x_{i-1}}{\Delta x_i}f(x_i) \leq M_i \end{align} if $t \in [x_{i-1}, x_i]$, i.e. you linearly interpolated the points. Observe $g$ is continuous on $[-\pi, \pi]$.