2

An odd function of period 2$\pi$ is appoximated by a Fourier Series with N terms. The appoximate error as measured by mean-square deviation is

$$E_N =\int\limits_{-\pi}^\pi\left( f(x) - \sum_{n=1}^N b_n \sin nx \right)^2 dx$$

By differentiating $E_N$ with respect to the coefficients $b_n$, find the values of $b_n$ that minimize $E_N$.

The main problem for me was I didn't know how to differentiate this function. Any help will be appreciated. Thanks.

DonAntonio
  • 211,718
  • 17
  • 136
  • 287
math
  • 1,509
  • 1
    Don't you have the definition of the coefficients $,b_n,$ ? These are functions of $,x,$ ... – DonAntonio Apr 11 '13 at 14:03
  • 1
    How can I differentiate with respect to funtion? – math Apr 11 '13 at 14:12
  • 1
    $$b_n=\frac{1}{\pi}\int\limits_{-\pi}^\pi f(x)\sin nx,dx \ldots$$ But perhps I see a problem here (unless you were given something different): the coefficients $,b_n,$ are not actually just functions of $,x,$ but in fact just numbers and thus their derivative is zero... – DonAntonio Apr 11 '13 at 14:17
  • 1
    I know this but it does not solve my problem. – math Apr 11 '13 at 14:19
  • Well, if the derivative is zero then you already know their derivative, don't you? :) – DonAntonio Apr 11 '13 at 14:20
  • I think the meaning of the question is the following: the coefficients $,b_n,$ themselves are the variables! So now we can derivative wrt them...Give it a try. – DonAntonio Apr 11 '13 at 14:21
  • 2
    I tried but failed, as I said while I ask this question. Please help DonAntonio. – math Apr 11 '13 at 14:22

1 Answers1

2

An idea only: putting $\,b:=(b_1,b_2,...,b_N)\,$ and regarding the given function as a real one on $\,\Bbb R^N\,$ :

$$\frac{d}{db}(E_N)=\frac{d}{db}\left[\int\limits_{-\pi}^\pi \left(f(x)-\sum_{n=1}^N b_n\sin nx\right)^2dx\right]=\sum_{k=1}^N\int\limits_{-\pi}^\pi\frac{d}{db_k}\left(f(x)-\sum_{n=1}^Nb_n\sin nx\right)^2dx$$

Assuming differentiation under the integral sign is allowed (Leibnitz theorem)

DonAntonio
  • 211,718
  • 17
  • 136
  • 287