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Consider $C([a,b], \mathbb{C})$, the space of continuous complex-valued functions on $[a,b]$. Define the complex inner product on $C([a,b], \mathbb{C})$ as $$\langle f,g\rangle=\int_a^bf(t)\overline{g(t)} \, dt .$$

Take $[a,b] = [0, 2 \pi]$. Show that all the functions in the set $\{ f_n := e^{inx}\mid n \in \mathbb{Z} \}$ are orthogonal. Also, let $W_n$ be the complex linear subspace spanned by $\{ f_m : |m| \leq n \}$. Compute the orthogonal projection $P_{W_n}$.

So proving that the functions are orthogonal was fine. However, I think I haven't completely solved the second part of the question. I know that $$P_{W_n}(f) = \sum_{i = -n}^n \frac{\langle f,f_i\rangle}{\langle f_i,f_i\rangle}f_i. $$ Furthermore, $\langle f_m,f_m\rangle = \int_{0}^{2\pi} e^{imx}e^{-imx}dx =\int_0^{2\pi}dx = 2\pi.$

Is this all there is to show? Because I am expecting some nicer result. Thanks in advance.

Michael Hardy
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Jeroen
  • 1,653
  • You can get an ONB from that, the coefficients of the projection will be the Fourier coefficients of you function... There's a lot of potential in this. – AlexR Sep 08 '13 at 21:03
  • Your formula for $P_{W_n}$ is fine, but write it down explicitly in terms of integrals and $f_i$'s and then simplify it – user8268 Sep 08 '13 at 21:07

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