what is beyond distribution theory ?? we have a definition of function $ y= f(x) $
we have a definition of generalized functions for example $ n^{a} f(nx) \to \delta (x) $
but what is 'beyond' the theory of distribution ?
in many cases you can not define a solution of a ODE in terms of a function
and you need to use distributions but is something beyond distribution theory? for example a distribution $ S(n,x) $ that whenever $ x \to \infty $ you have some new mathematical object
 A: This is a way to study a major subject in analysis, when we have a function sequence $f_n$ and
$$\forall \varphi \in X, \qquad \lim_{n \to \infty} \int_{-\infty}^\infty f_n(x) \varphi(x)dx \quad \text{is well-defined}$$
Where $X$ is some set of functions ($L^1,L^2,C^0,C^\infty_c, S(\mathbb{R})...$, you need to know some functional analysis, Banach and Hilbert spaces, dual, topological vector spaces, operator norm)
Taking $f_n(x) = 2n\, 1_{|x| < 1/n}$ then the limit is well-defined whenever $\varphi $ is continuous at $x=0$. We call $\delta(x)$ this limit and write
$$\int_{-\infty}^\infty \delta(x) \varphi(x)dx \overset{def}= \lim_{n \to \infty} \int_{-\infty}^\infty 2n\, 1_{|x| < 1/n} \varphi(x)dx$$
When choosing $X = C^0$ we get that such limits are well-defined whenever $f_n$ converges in the dual of $C_0$ : the distributions of order $0$.
When choosing to $X=  C^\infty_c$ we obtain that such limits are well-defined whenever $f_n$ converges "in the sense of distributions".
This set is interesting because it is closed under differentiation (when $f_n$ is differentiable then $\int_{-\infty}^\infty f_n(x) \varphi'(x)dx =- \int_{-\infty}^\infty f_n'(x) \varphi(x)dx$ and this property extends to the limit).
We can choose a slightly larger set for $X$, the Schwartz space, which is interesting because it is closed under the Fourier transform, and this property extends to its dual, the tempered distributions.
From a distribution $T$ you can get a sequence $f_n \in C^\infty_c$ that converges to it in the sense of distributions : $f_n(x) = \phi_n(x) (\varphi_n \ast T(x))$ where $\phi_n \in C^\infty_c, \phi_n \to 1$ in $C^0$ and $\varphi_n \to \delta$.
