Firstly - I am not working in the maths field, (so please be gentle), but I find maths interesting and would like to learn more while the brain-cells still permit.
I was posed a question recently which I need assistance with. It relates to the Discrete Fourier Transform and the calculation of the Root Mean Square values of both the original samples and the transform result.
BTW I am not at school, so this is not homework!
if $F$ is the Discrete Fourier Transform of a sequence of values $X_N$ length $N$ and $RMS$ is the Root Mean Square of a sequence of values length $N$, how can one prove (or disprove) that ...
$RMS( X_N)=RMS(F(X_N))$
I understand some of the principles of Fourier Transformation and the relationship with primitive roots of unity, but not on a really detailed level.
Also I am keen to learn the correct mathematical way to denote this and to approach the problem.
Expressed as linear combination in any orthonormal basis, the square length of a vector is the sum of squares of its coefficients. To be more precise, let $\langle \cdot, \cdot \rangle$ be a (hermitean) inner product on the $n$-dimensional vector space $V$ and let $v_1, \ldots, v_n$ be any orthonormal basis of $V$. That is, $$\langle v_k, v_m \rangle = \delta_{km}$$ for any indices $k$, $m$. Then for a vector $v = c_1v_1+\ldots+c_nv_n$ expressed in this basis $$\lVert v \rVert^2 = \langle v,v\rangle = \lvert c_1\rvert^2 + \ldots + \lvert c_n\rvert^2.$$ In this light, your observation comes down to the fact that the standard basis $e_1, \ldots, e_n$ and the Fourier basis $\omega_1, \ldots, \omega_n$ are both orthonormal. Here $\omega_k$ is the vector $$\omega_k = \frac1{\sqrt{n}}(1, \zeta^k, \zeta^{2k}, \ldots, \zeta^{(n-1)k})$$ with $\zeta$ a primitive $n$-th root of unity (usually $\exp(2 \pi \mathrm{i}/n)$).