Can Dirac Delta Function Test Functions Be Less Restrictive Than Schwartz Space?

Question

I understand that, in $L^2$ Hilbert space, functions do not exist pointwise, so if function $\phi(x) \in L^2$, we cannot speak of a specific value $\phi (0)$, at $x=0$, and so we cannot say $\int_{- \infty}^{\infty} \phi(x) \delta (x) dx = \phi (0)$.

But if we look at the space of functions, $\psi(x)$ (with $x \in R$), that both exist pointwise (so that $\psi(x)$ has a specifically defined value for any $-\infty \leq x \leq \infty$) and are square integrable from $-\infty$ to $\infty$, then does it make sense to let $\delta (x)$ act on those such that we can say: $\int_{- \infty}^{\infty} \psi(x) \delta (x) dx = \psi (0)$? I know that this simple definition of $\psi(x)$ does not guarantee the existence of the derivative of the delta function in this space but, that not withstanding, I think that $\int_{- \infty}^{\infty} \psi(x) \delta (x) dx = \psi (0)$ would still be true for those $\psi(x)$, especially if we define $$\int_{- \infty}^{\infty} \psi(x) \delta (x) dx \equiv \lim_{n\to\infty} \int_{- \infty}^{\infty} \psi(x) \delta_n (x) dx$$ with something like $\delta_n (x) \equiv \sqrt{\frac{n}{\pi}} \exp (-nx^2)$

What I am getting at is that it seems to me that the space of test functions for $\delta (x)$ need not be something as restrictive as Schwartz (S) or Kantorovich (K) space.

Answer 1

Your example fails; consider $\psi$ given by

$$ \psi(x) = \begin{cases} 1 & x = 0 \\ 0 & x \neq 0 \end{cases} $$

$\psi$ is square integrable,

$$ \psi(0) \neq 0 = \lim_{n \to \infty} \int_{-\infty}^{\infty} \psi(x) \delta_n(x) \, \mathrm{d} x $$

so in the dual space to square-integrable functions, $\delta \neq \lim_{n \to \infty} \delta_n$.

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The most reasonable general space to find delta distributions is, in my opinion, in the dual space to the space of continuous functions, if you simply want them to exist.

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But you should pay attention to the fact that, usually, the point isn't simply to define delta distributions — instead one wants to define a space that:

• Contains the objects of interest
• Admits all of the sorts of calculations we want in our application
• Has good analytical properties

The second point is very important. For example, in physics, Fourier transforms are very important, which makes the space of tempered distributions (on Schwartz functions) a popular one to work within.