I am considering the inner product space $L_{2 \pi}^2$ of square integrable 2$\pi$-periodic functions on $\Re$, with the inner product defined by: $<f,g>$ = $\int_{0}^{2\pi} f(x)g(x) dx$.
For $f$ $\in$ $L_{2 \pi}^2$, let $S_nf$ be the best $L^2$-approximation to $f$ out of $\prod_n$, where $\prod_n$ is the space of all trigonometric polynomials of degree at most $n$.
Show that for every $f$ $\in$ $L_{2 \pi}^2$, $||S_nf||_2 \leq ||f||_2$.
I have worked out and have the following:
$S_nf(x)$ = $\frac{a_0}{2} + \sum_{k=1}^{n} (a_k cos kx + b_k sin kx)$ and $\int_{0}^{2\pi} |\frac{a_0}{2} + \sum_{k=1}^{n} (a_k cos kx + b_k sin kx)|^2 dx \leq \int_{0}^{2\pi} |f(x)|^2 dx$.
By Bessel's Inequality: $||f||_2^2 \geq |<f,\frac{1}{\sqrt{2\pi}}>|^2 + \sum_{k=1}^{n} (|f,\frac{cosk\pi}{\sqrt{\pi}}|^2 + |f,\frac{sink\pi}{\sqrt{\pi}}|^2 ) $ After some simplifcation: $||f||_2 \geq (\frac{\pi a_o}{2})^{0.5}+(\sum_{k=1}^{n} (\pi (a_k^2+b_k^2))^{0.5}$
However, I do not know how to proceed. Thank you for your help!
Note: we define $n^{th}$ partial sum $S_nf$ of the Fourier Series of $f$