My text book, "continuous and discrete signals and systems 2/e by Soliman and Srinath, specifies sufficient conditions of convolution integral.
$$y(t) = x(t) * h(t) = \displaystyle \int_{-\infty}^{\infty}x(\tau)h(t-\tau)d\tau,$$
Sufficient conditions for existence of y(t) are:
- two functions $x(t)$ and $h(t)$ should be absolute integrable, $t\in(-\infty, 0]$
- two functions $x(t)$ and $h(t)$ should be absolute integrable, $t\in[0, +\infty)$
- At least one function among $x(t)$ and $h(t)$ should be absolute integrable, $t\in(-\infty, +\infty)$
I don't understand why the text book says the same thing over again.
I consider condition 1 + 2 = conditoin 3.
Are there any functions which is absolute integrable $t\in(-\infty, 0]$ and $t\in[0, +\infty)$ but not integrable in $t\in(-\infty, +\infty)$.
Why the textbook express like this? Please let me know, Thank you!