How is it justified to apply the $\delta$ function to functions without compact support

distribution-theoryfunctional-analysis

The $\delta$ function is defined as a distribution, and is thus an element of $\mathcal{D}'(U)$ for some open set $U \in \mathbb{R}^n$. In other words, it is a function in a dual space, and hence we may write $\delta: C_c^\infty(U) \rightarrow \mathbb{R}$ where $C_c^\infty(U) = \mathcal{D}(U)$ is the set of smooth functions over $U$ with compact support.

What has been bothering me is that I often see expressions such as
$$\int \delta(x)e^x dx.$$
But the function $e^x$ does not have compact support over any open set. Likewise, but less common, I also have seen the $\delta$ function applied to functions that are not infinitely differentiable.

How are such uses of the delta function (and distributions in general) justified?

Best Answer

Given any sets $A,B$, and any point $a\in A$, I can always decide that I want to define the function $\delta_a:F=\text{Functions}(A,B)\to B$, $\delta_a(f):= f(a)$. I can call this an evaluation at $a$ mapping, or I can call this the Dirac delta centered at $a$, or I can give it any other name I like. Point is, I can define it because it makes sense. The question is if this is useful?

Well, Dirac deltas are mostly useful in vector spaces, so it would be a good idea to consider the target $B$ to be a vector space. Also, we want to deal with this in analysis, so the most common choice would be $A$ an open set in $\Bbb{R}^n$, and $B=\Bbb{C}$ (but again, if you're going to be fancier, you may want to allow the domain $A$ to be something else, and you may want to allow $B$ to be any Banach space). Also, for the purposes of analysis, I usually don't just want to consider all functions $A=U\to\Bbb{C}$, I probably only want to consider some of them, and I probably want to equip a topology on these function spaces. Some common choices are

  • $A=U$ open in $\Bbb{R}^n$, $B=\Bbb{C}$, $F=C_b(U)$, the bounded complex-valued continuous functions on $A$ with say the supremum norm, and $\delta_a:C_b(U)\to\Bbb{C}$ defined as above. Then, you can show $\delta_a$ is continuous and linear (bounded continuous functions is an important space, e.g in probability, and also in many areas of analysis; look up the Riesz representation theorem for measures and integrals).
  • You can consider $A=U$ open in $\Bbb{R}^n$, $B=\Bbb{C}$, $F=\mathcal{E}(U)=C^{\infty}(U)$, the space of smooth functions equipped with say the topology of uniform convergence on compact subsets of all derivatives (so giving you a locally convex topological vector space rather than a Banach space). You can again consider $\delta_a:\mathcal{E}(U)\to\Bbb{C}$, defined as above. You can show that it is again continuous and linear with respect to this topology. In fact, continuous linear maps on this larger space of functions can be identified with distributions of compact support (see for example Folland's real analysis text for more information).
  • You can also consider it as a mapping $\delta_a:\mathcal{S}(\Bbb{R}^n)\to\Bbb{C}$, as a linear mapping on Schwartz functions. You can show it is continuous and linear here as well... the name given here is that it is a tempered distribution, i.e an element of $\mathcal{S}'(\Bbb{R}^n)$ (so the theory of Fourier transform applies here).
  • You can also consider it as a mapping $\delta_a:\mathcal{D}(U)\to\Bbb{C}$, where it is again continuous and linear... i.e a distribution in the usual sense of Schwartz.

The last three are reflecting the idea that, on $\Bbb{R}^n$, $\mathcal{D}\subset \mathcal{S}\subset \mathcal{E}$ (in the sense of there is a natural inclusion, which is continuous in the appropriate sense). So their topological duals satisfy $\mathcal{E}'\subset\mathcal{S}'\subset\mathcal{D}'$.

There are of course a whole bunch of other spaces on which you can consider things (e.g manifolds, vector bundles blablabla). The question though is for what purpose are you using it. Beyond a certain point it becomes pretty irrelevant whether you want to call it a distribution, or a functional or a linear form, or whatever, because people will freely interchange terms. One has to use context to decipher the intended meaning.

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