Defining the value of a distribution at a point

As indicated above, the concept of the limit resp. value of a distribution at a point was studied intensively over 50 years ago. Here is a very elementary and natural definition due to Sebastião e Silva (it is definition 6.9 in his paper „On integrals and orders of growth of distributions“. (I will not give a reference since it can be found online just by googling the title).

A distribution $s$ on an interval $I$ is said to be continuous at a point $c$ if there is a natural number $p$ and a continuous function $F$ on $I$ so that $f=D^pF$ (distributional derivative) and $\dfrac {F(x)}{(x-c)^p}$ converges in the usual sense as $x$ goes to $c$ Then we write $f(c)$ for $p!$ times this limit and call it the value of the distribution at $c$. As an example, he shows that $\cos \frac 1 x$ has the value $0$ at $0$.

It's not a bad definition and I think it is better to think of it as a particular case of the "restriction problem", i.e., trying to define the restriction $\omega|_{\Gamma}$ of $\omega$ to some subset $\Gamma\subset\mathbb{R}^n$. When one succeeds the result is called a trace theorem. This usually requires some quantitative regularity hypothesis on $\omega$, e.g., being in a Sobolev space $H^s$ with $s>$something.

A particularly important case is when $\Gamma$ is an affine subspace, or say for simplicity a linear subspace like $\Gamma =\mathbb{R}^m\times\{0\}^{n-m}\subset\mathbb{R}^n$. A rather standard approach is to start with $\omega\in\mathcal{D}'(\mathbb{R}^n)$. The convolution $\omega\ast \phi_j^0$ is in the space of $C^{\infty}$ functions $\mathcal{E}(\mathbb{R}^n)\subset \mathcal{D}'(\mathbb{R}^n)$ and converges to $\omega$ in the topology of $\mathcal{D}'(\mathbb{R}^n)$ (the strong topology). The ordinary restriction $\omega\ast \phi_j^0|_{\Gamma}$ makes sense and you can ask if the limit $\lim_{j\rightarrow\infty}\omega\ast \phi_j^0|_{\Gamma}$ exists inside $\mathcal{D}'(\mathbb{R}^m)$.

Your particular case $p=0$ corresponds to mine with $m=0$.

Another problem of this kind is pointwise multiplication. If $\omega_1(x)$ and $\omega_2(x)$ are two distributions, then there is no problem defining $\omega_1(x_1)\omega_2(x_2)$ (tensor product), but the issue is how to restrict to the diagonal $\Gamma=\{x_1=x_2\}$.

Finally, note that all of these problems become much more interesting for random distributions, because it's like magic: you can sometimes do the (deterministically) impossible.

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Small addendum: Suppose that for some reason one has a trace theorem but only for large enough $m$ and one cannot do the $m=0$ or the point restriction case. Then one can still do the following "stabilization" trick: change $\omega$ to $\omega\otimes 1$ where one tensors with the constant function equal to one seen as a distribution in say $p$ new variables. If you can restrict it from $\mathbb{R}^{n+p}$ to a subspace of dimension $p$, then you will have your point evaluation after factoring out the $\otimes 1$. The last step of course needs your restriction construction to be invariant/covariant by translation along $\Gamma$.

The definition of the value of a distribution at a point you describe in your question does not seem flawed to me since, at least from the point of view of the independence on $\delta$-sequences, follows the path traced years ago by Stanisław Łojasiewicz in the paper [1], so I describe his approach to the problem below.

Łojasiewicz analyzes the problem for functions of one variable, i.e. $n=1$: by using the definition of change of variables in a distribution (see for example [2], §1.9 pp. 21-22) and considering the change of variable $y=x_0+\lambda x$, for $ x,x_0,\lambda \in\Bbb R$, i.e. $$ \begin{split} T(x_0+\lambda x)&\triangleq \langle T(x_0+\lambda x),\varphi(x)\rangle\\ &=\left\langle T(y),\frac{\varphi\big(\lambda^{-1} (y-x_0)\big)}{\lambda}\right\rangle \end{split} \quad \varphi\in\mathscr{D}(\Bbb R) $$ he defines the limit of a distribution at a point $x_o$ as ([1], §1 p. 2-3) $$ \lim_{x\to x_0} T\triangleq \lim_{\lambda\to 0} T(x_0+\lambda x) \label{1}\tag{1} $$ and proves that

• $\lim_{x\to x_0} T=\lim_{x\to x_0^+} T=\lim_{x\to x_0^-} T$
• by using an earlier result of Zieleźny, if the limit \eqref{1} exists, it is necessarily a constant $C\in \Bbb C$, or more precisely a constant distribution $C$.
• a necessary and sufficient condition for the limit \eqref{1} to exist is (see [1], §2, theorem 2.2, pp. 5-7) that $T=f^{(n)}$, where $f\in C^0(\Bbb R)$ and $$ \lim_{x\to x_0}\frac{f(x)}{(x-x_0)^n}=\frac{C}{n!}. $$

Then Łojasiewicz assumes \eqref{1} as the definition of the value of a distribution at a point: note that this definition does not rely on any particular test function (or sequence of such) $\varphi\in\mathscr{D}(\Bbb R)$, as stated above. Now a few observations:

1. Łojasiewicz ([1], §1 p. 1) states that the case $n>1$ will be analyzed in a subsequent paper which to my knowledge has never been published. However (but this only my opinion), a generalization of \eqref{1} could perhaps be tried by using the Stoltz condition as described, for example, in the textbook of Griffith Bailey Price (1984) Multivariable Analysis, Springer-Verlag.
2. Łojasiewicz gives another necessary and sufficient condition for the limit \eqref{1} to exists, in terms of Denjoy differentials ([1], §2, corollary to theorem 2.2, p. 7).
3. The term $\lambda^{-1}$, more or less intrinsically used in \eqref{1}, suggests the possible use of the Mellin transform: this suggestion was followed by Bogdan Ziemian in [3], §12 pp. 41-42. He defines a (generalized) spectral value of a function/distributions at a point and proves ([3], §12 p. 43) that it coincides with Łojasiewicz point value \eqref{1} when this exists (by using the necessary and sufficient condition above): the construction of Ziemaian however does not apply to all distributions.

[1] Stanisław Łojasiewicz (1957-1958), "Sur la valeur et la limite d'une distribution en un point" (French), Studia Mathematica, Vol. 16, Issue 1, pp. 1-36, MR0087905 Zbl 0086.09405.

[2] V. S. Vladimirov (2002), Methods of the theory of generalized functions, Analytical Methods and Special Functions, Vol. 6, London–New York: Taylor & Francis, pp. XII+353, ISBN 0-415-27356-0, MR2012831, Zbl 1078.46029.

[3] Bogdan Ziemian (1988), "Taylor formula for distributions", Rozprawy Matematyczne 264, pp. 56, ISBN 83-01-07898-7, ISSN 0012-3862, MR0931848, Zbl 0685.46025.