I am reading a book on signal processing, rigor level of which is lower than a thorough introduction to functional analysis and is higher than an engineering introduction of signals and systems. The biggest difference of the material, from engineering signal processing books, is to explain the space of tempered distributions using Schwartz functions and to explain how $\delta$ is defined a tempered distribution, and to introduce Fourier transforms on the space of tempered distributions.
I am half way through the book, and I have some questions to ask. The biggest question I have, is how to view previous engineering writing. For instance, we know that $\delta$ is not a function, but a distribution. We are used to seeing equations of the following style: \begin{equation} \delta * f = f, \end{equation} and \begin{equation} \delta(x-a) * f\left(x\right) = f\left(x - a\right). \end{equation}
My current thought is that, these writings are abbreviations for the following steps: 1. do convolution in functional space 2. convert the result back to function space (if there is a preimage). In this case, there is no need to raise $f$ to a tempered distribution, as a function convolving a distribution is well-defined. In general, my understanding is that, whenever some operation in an engineering signals book is invalid in the function space (no dirac delta, Fourier transform doesn't exist, etc), raise that to the functional space, and use the corresponding operation in the functional space to find a distribution. If the distribution is of a preimage in the function space, convert it back for engineering understanding. Of course, previous conclusions are still valid, but the true theoretical thoughts behind them are more complicated than they look. Is this understanding correct?