Transform pair

Remarks

Example: for the Laplace transform $T$ and a real number $a$, the pair $(e^{at}, 1/(s - a))$ is a transform pair, because $\int_0^\infty e^{at} e^{-st}\, dt = 1/(s - a)$ for $s > a$. Non-example: $(1, 1/s^2)$ is not a Laplace transform pair, because the Laplace transform of $1$ is $1/s$, not $1/s^2$. When $T$ is injective on $A$, each $F$ in the image determines $f$, and the pair can be read in both directions.