It's hard to tell from the picture but it looks like you have mainly even harmonics.
Even and odd harmonics have fairly different root causes. Loosely speaking, harmonics are caused by a non-linear input output relationship somewhere in your system. Even harmonics are caused by asymmetries, i.e. $y(x) \neq -y(-x)$. Odd harmonics are caused by the relation ...
You have guessed it right. But before an explanation, you should make sure that your measuring setup is not the cause of this observation.
The fact that your fundamental frequency is 0.7 Hz. does not mean that you should see all harmonics of that frequency at 1.4 Hz., 2.1. Hz, 2.8 Hz. etc...
And furthermore, even if you would see those harmonics, their ...
The discrete-time Fourier transform (DTFT) can be used for general infinite length signals, and it can also be used for periodic signals if we allow Dirac delta impulses in the expression for the DTFT. You have the correspondence
where it is understood that the expression $(1)$ is valid ...
You're thinking too complicated. Sorry I wasn't clear enough in my comment.
You're given a sequence $x[n]$ that contains three spikes and zeros elsewhere. Your sequence $y[n]$ contains periodically shifted copies of the original one. It's just repeating the same three spikes indefinitely, without overlap since $7>3$. Hence, all you need to do to find out ...
Thanks to @Florian's comments, here is sample solution
n = 0:1:7*10^6-1;
x = zeros(size(n));
x( find(n==1 | n==2 | n==3) ) = [(-1)^1*1 (-1)^2*2 (-1)^3*3];
y = x(mod(n,7)+1);
y = sum(reshape(y,7,),2)'
For large $k$s we have for example: y = 0 -1000000 2000000 -3000000 0 0 0.