time downsampling vs. frequency downsampling $x[n]_M$ is a finite length sequence of length M.

if:

$$
y = x[nN]_M    \tag{1}
$$

is called downsampling in the time-domain.

then what do you call the process of converting going from a M-point to an N-point DFT?

$$
y[n] = \Bigg[ \sum_{k=-\infty}^{\infty} x[n-Nk]_M \Bigg] R_N[n] \tag{2}
$$

does that have a name?  like frequency downsampling? frequency rebinning...frequency re-pie cutting or something like that?
just guessing...

($R_N[n]$ = window function, which is 1 between 0 and N-1 and 0 otherwise.)

$$
Y[k]_N = X(e^{j\omega})\bigg|_{\omega=2\pi k/N} \tag{3}
$$

rebinning $Y[k]_N$ to $Y[k]_M$ for example.

>
Your second equation suggests that $y[n]$ is obtained by performing an $N$-point inverse DFT of an $M$-point DFT $X[k]$ of an $M$-point sequence $x[n]$.

The result can be interpreted as shifted copies of $x[n]$ are superimposed, causing possible aliasing in time, which depends on whether $ N \geq M$ or $N &lt; M$.

If $N \geq M$, there is no aliasing and $y[n]$ will be an $N$ point sequence whose first $M$ samples will be identical to $x[n]$ for $n=0,1,...,M-1$ and remaining $N-M$ samples will be zero-padded.

$$
y[n] = \begin{cases} { x[n] ~~~,~~~ n = 0,1,...,M-1 \\ ~~~~~0 ~~~,~~~ n= M,M+1,...N-1} \end{cases}
$$ 

Otherwise if $N &lt; M$, then $y[n]$ will be a time-aliased version of $x[n]$, as shifted copies of $x[n]$ would be overlapping each other. If you want any non-aliased samples in $y[n]$, then $N &gt; M/2$ in general and in such a case $y[n]$ could be defined as:

$$
y[n] = \begin{cases} { x[n]+x[n+N] ~~~, n = 0,1,...,M-N \\ x[n] ~~~~~~~~~~~~~~~~~~~~~,~~~ n= M-N+1,...,N-1 } \end{cases}
$$