Mathematical terms for "bandlimited" and "timelimited"? I have read

"Signals that are bandlimited are not timelimited" and the reverse; "Signals that are timelimited are not bandlimited".

Q1: Is this because of the Fourier transform?
Q2: What are the mathematical terms for "bandlimited" and "timelimited"?
 A: A signal $x(t)$ is time limited if there exists a $T>0$ such that $$x(t)=0\qquad,\qquad |t|>T$$so we can define band limited (in frequency) signals where their FT is zero for $\omega>\omega_0$ and for some $\omega_0$. Also a signal limited on one domain (time or frequency) cannot be limited in the other domain either, but there are signals neither limited in time nor in frequency for example consider $x(t)=e^{-|t|}$ with corresponding FT $X(\omega)=\dfrac{2}{1+\omega^2}$. Neither $x(t)$ nor $X(\omega)$ are limited in time and frequency respectively. To show that, let $x_T(t)$ be a time limited version of signal $x(t)$ in $|t|<T$ therefore$$x_T(t)=x(t)\Pi(\dfrac{t}{2T})$$by taking FT we have$$X_T(\omega)=X(\omega)*2Tsinc(\dfrac{T\omega}{\pi})=$$regardless of $X(\omega)$ being band limited or not, $X_T(\omega)$ is never band limited because of the convolution of $X(\omega)$ with $sinc$ function and this is what we wanted to show. The same argument can be used in dual case for band limited signals.
