Weird change of variable in summation? I don't understand the change of variable in this summation
$$
\frac{1}{T}\sum_{k=-\infty}^{\infty} H \Big (\frac{F-k}{T}\Big )=1
$$
Change of variable: $f=\frac{F}{T}$ so
$$
\frac{1}{T}\sum_{k=-\infty}^{\infty} H \Big (f-\frac{k}{T}\Big )=1
$$
Isn't this wrong? $T$ isn't changed?
Shouldn't it instead be
$$
\frac{f}{F}\sum_{k=-\infty}^{\infty} H \Big (f-\frac{f}{F}k\Big )=1 \quad \text{?}
$$

In fact, both forms are correct. However when solving certain problems, not substituting the change for all variables is convenient.
You have your summation:
$$\frac{1}{T}\sum_{k=-\infty}^{\infty} H \Big (\frac{F-k}{T}\Big )=1$$
The fractions may be separated to give:
$$\frac{1}{T}\sum_{k=-\infty}^{\infty} H \Big (\frac{F}{T}-\frac{k}{T}\Big )=1$$
And so now, you can apply the substitution $f=\frac{F}{T}$ for only the $\frac{F}{T}$ term:
$$\boxed{\frac{1}{T}\sum_{k=-\infty}^{\infty} H \Big (f-\frac{k}{T}\Big )=1}$$
Note that: $$f=\frac{F}{T} \Rightarrow T=\frac{F}{f}$$
Therefore, you can substitute for $T$ to give your version of the summation, where you do not have the summation in terms of $T$ at all.
$$\frac{1}{(\frac{F}{f})}\sum_{k=-\infty}^{\infty} H \Big (f-\frac{k}{(\frac{F}{f})}\Big )=1$$
$$\boxed{\frac{f}{F}\sum_{k=-\infty}^{\infty} H \Big (f-\frac{f}{F} k\Big )=1}$$
