Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

We know that the Fourier system is complete, i.e. that $\lbrace e_n: ~ n \in \mathbb{N} \rbrace$ defined by \begin{equation} e_n(x)=\frac{1}{\sqrt{2 \pi}}\exp(inx), ~~~ n \in \mathbb{Z} \end{equation} is an orthonormal basis of $L^2(-\pi,\pi)$.

My question is:

How do you proof, that \begin{equation} e_n(x)=\frac{1}{\sqrt{2N}}\exp\left( \frac{i \pi n}{N}x \right), ~~~ n \in \mathbb{Z} \end{equation} defines an orthonormal basis on $L^2(-N,N)$ (if it does)?

share|cite|improve this question
up vote 3 down vote accepted

If your first formula is correct, your second can't be. Taking $N=\pi$ you don't have normed vectors. After scaling them the proof should be the same with a little substituion in the Integrals. Edit: you corrected the formel, now just using the subsitution rules for Integrals will bring you the result.

share|cite|improve this answer
Thank you. So I will have to do the proof again. There is no way to proof this when the basis for $L^2(-\pi,\pi)$ is already known? – mjb Feb 15 '13 at 13:51
With the subsitution in the Integral, you have the orthonormality, and in addition the basis. – Dominic Michaelis Feb 15 '13 at 13:54

In general, for a proposed family $\{e_n(x)\}$, to determine if it is orthonormal, just check whether $$\langle e_n(x),e_m(x)\rangle=\begin{cases}1, &n=m,\\ 0, &n\not=m,\end{cases}$$ where here the inner product is $$\langle u,v\rangle=\int_a^b u(x)\overline{v(x)}\,dx.$$

To show that an orthonormal family forms an orthonormal basis, you need to show that the family is complete. Two criteria which are equivalent in your setting are:

  1. $\langle f,e_n\rangle=0$ for all $n$ implies $f$ is the zero function.
  2. Parseval's equation is true for all $f\in L^2[a,b]$: $\|f\|^2=\sum_n c_n^2$ where $c_n$'s are the coefficients in the expansion of $f$ in this orthogonal family.
share|cite|improve this answer
Yes, but this is just a proof for orthonormality and not for the basis-assumption – mjb Feb 15 '13 at 13:50
Thank you! But this how do I actually do that? I know the equivalent characterizations of bases, but which one to use and how? – mjb Feb 20 '13 at 6:50

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.