Generalized function theory bothers me from time to time, I used to put aside and not delve into it, but this time I encountered it again with annoyance and wanted to understand it to some extent. Below is my current core questions.
Generalized functions are continuous linear functionals defined on test functions space with compact support, which have two notations, symbolic(eg (x)) and functional(eg [φ]). The pdf(https://www.cs.odu.edu/~mln/ltrs-pdfs/tp3428.pdf) I have read do not elaborate on this point further, and I have the following questions:
Can any given continuous linear functional on D give a symbolic notation of it? If yes, how to give? What is the general bridge between the symbolic notation and the functional notation?
For common operations such as multiplication, differentiation, limit, integral, etc. what are the correspondences/relations between these two notation-systems? For example, if I derive the symbolic (functional) form, what is derivatives of the functional (symbolic) form?
why generalized function always equal to the generalized derivatives of an ordinary function? can you give me some easy examples to explain it?
I'm not a math-majored student, so I'm not greedy to understand from underlying, I just want to master operations of generalized functions and understand it from relations of the two notation-systems.
The theory of distributions generalizes the concept of functions and operations theoron, e.g., differentiation, and composition, to more "rough" objects (hence the other name of the theory, i.e., the theory of generalized functions). Since many operations on functions can be naturally extended to distributions, it is tempting to use the symbolic notation for distributions, i.e., that of functions.
No. Only for some distributions the symbolic notation actually makes sense mathematically. The minimum "regularity" you need on the distribution for the functional notation to actually correspond to the symbolic one is that of local integrability. In other words, given a test function $\phi$ and a distribution $u$, the action of the distribution on the test function $u[\phi]$ corresponds to the integration $\int u(x) \phi(x) dx$ if $u \in L^1_{\text{loc}}$, i.e., if $u$ is at least a locally integrable function, where here the integration is in the sense of Lebesgue. In such a case it is mathematically valid to talk about an object such as $u(x)$. When the distribution is more "rough" than a locally inegrable function, like a dirac delta, a symbolic representation such as $\int \delta(x) \phi(x) dx$ does not have any mathematical meaning, i.e., it does not represent a Lebesgue integral, but it seems to still be a useful abuse of notation. Indeed, such a symbolic notation is used in physics literature whether or not the distribution has enough regularity.
Operations on distributions are defined in such a way as to mimic operations on locally integrable functions, and such that the resulting objects are distributions. That's why, a direct correspondence between the two formulations can be established. Let's take an example. Given a distribution $u$, its derivative $u'$ is defined by its action on a test function $\phi$ as:
$$ u'[\phi] = -u[D \phi], $$ where $D\phi$ is the (normal) functional derivative of $\phi$. That is, $u'$ is the distribution that acts on $\phi$ the same way that the distribution $u$ acts on the derivative of $\phi$ with another sign. Assume now that both distributions $u$ and $u'$ are both locally integrable functions, then the functional notation actually corresponds exactly to the Lebesgue integral: $$ \int u'(x)\phi(x) \ dx = - \int u(x) \phi'(x) \ dx. $$ Notice that, to go from the left-hand to the right-hand side of the last equation you actually use integration by parts. Hence, the derivative of a distribution is actually defined in such a way as to mimic the integration by parts formula for normal Lebesgue integrals. Now if $u$ and $u'$ are less regular than locally integrable functions, the symbolic notation is not anymore meaningful, but people still use it since it is a convenient abuse of notation.
Finally, I try to address your confusion on the relationship between generalized derivatives of functions and relationship to distributions. Let's take for that the example you posed in the comments, i.e., $H' = \delta$. Note that the Heaviside function has a jump at zero and, hence, there is no meaning for a derivative of this function in the standard sense of differentiation. However, we can find the weak (aka generalized) derivative of $H$. This is done by multiplying by a smooth and compactly supported function $\phi$, integrating and then doing integration by part.
$$ \int_{\mathbb{R}} H'(x) \phi(x) \ dx = - \int_{\mathbb{R}} H(x) \phi^{'}(x) \ dx = - \int_{0}^{\infty} \phi^{'}(x) \ dx = -\phi(x)|_{0}^{\infty} = \phi(0) $$
Note that, by definition, $\delta [\phi] = \phi(0)$. And we can write $\int_{\mathbb{R}} H'(x) \phi(x) \ dx = H'[\phi]$ (functional notation). Thus we have $H'[\phi] = \delta [\phi]$ for any test function $\phi$. In such a case we say that $H' = \delta$ in the sense of distributions. The idea in a nutshell is that many functions don't have derivatives (in the standard sense) because they are not regular enough. However, we can still find derivatives for these functions in the sense of distributions.