Eigenfunctions of discrete-time LTI systems
i think the only family of functions that are eigenfunctions to LTI systems the sole exponential.

for continuous-time, if it's LTI:

$$ y(t) = \int\limits_{-\infty}^{+\infty} h(u) \ x(t-u) \ du $$

the eigenfunction is $x(t) = A e^{st}$ and the output is

$$ y(t) = H(s) \ x(t) $$

where

$$ H(s) = \int\limits_{-\infty}^{+\infty} h(t) \ e^{-st} \ dt $$

for discrete-time LTI systems, it is similar

$$ y[n] = \sum\limits_{m=-\infty}^{+\infty} h[m] \ x[n-m] $$

the eigenfunction is $x[n] = A z^n$ and the output is

$$ y[n] = H(z) \ x[n] $$

where

$$ H(z) = \sum\limits_{n=-\infty}^{+\infty} h[n] \ z^{-n} $$

often $s=j\Omega$ or $z=e^{j\omega}$ but it wouldn't have to be that.