Complex Fourier transform (represented by Σ)

[arcTomato]

TL;DR Summary

I can't understand how to derive fourier transform(described by Σ).

Dear all.

I can't understand how to derive Eq.(2.3a).

Fourier coefficients, $A_j$ and $B_j$ are described by summation in this paper as (2.2).　I think this is weird.
Because this paper said "In this section 2.1 ,the Fourier transform is introduced in very general terms".
and I understand Fourier coefficients as integral.( I think integral version is more general than $Σ$ version.) so I'm confused.

I would like to ask you how to derive (2.3a) in this paper( in $Σ$ version).
(this is the paper I readFourier_techniques.pdf)

 

[tnich]

Assume that you are integrating from $-T$ to $+T$, and consider the evenness or oddness of $\cos^2\omega_jt$, $\sin^2\omega_jt$, and $\sin\omega_jt\cos\omega_jt$. Then take the small step from integration to discrete summation.

 

[arcTomato]

I appreciate for you, @tnich.

Can I use $t_k=kT/N,ω_j=2πj/T$in the process to derive (2.2)??
This paper does not say anything about $ω_j$ in section 2.1.
So you say like this??Can I approximate to Σ from integral?

 

[tnich]

Yes, I am saying that you can approximate the integrals with discrete summations. Try looking at the definition of an integral as a limit of discrete summations to see how to do the approximation.
I am having trouble reading your derivation, so I don't know if you got it right. I see that you kept $k$ and $t_k$ in the integral. I think you need to integrate over $t$ from $-T$ to $+T$.
But, that said, on page 2 the author says that this section is a general overview and that he will go into the details later. If you understand the general idea, I would be tempted to just read on.

 

[arcTomato]

Ok, I think I got it
Thanks for your kindness,@tnich!