The Calculus of Distributions

The key idea throughout this section is the same: if an operation $A$ on smooth functions satisfies $\langle Af, \varphi \rangle = \langle f, A^\dagger \varphi \rangle$ for some adjoint $A^\dagger$ that maps test functions to test functions, then we define $\langle AT, \varphi \rangle = \langle T, A^\dagger \varphi \rangle$ for any distribution $T$.

A note on notation. We have a functional $T : \mathcal{D}(\Omega) \to \mathbb{R}$, and we want to define a new functional $AT$. The definition always acts on the test function: $AT$ is the composition $T \circ A^\dagger$, and $T$ itself is unchanged. Concretely:

• $D^\alpha T = (-1)^{|\alpha|}\, T \circ D^\alpha$, i.e. $\varphi \mapsto (-1)^{|\alpha|} T(D^\alpha \varphi)$,

• $fT = T \circ M_f$, i.e. $\varphi \mapsto T(f\varphi)$, where $M_f$ is the multiplication operator,

• $\tau_h T = T \circ \tau_{-h}$, i.e. $\varphi \mapsto T(\tau_{-h}\varphi)$.

In each case the notation ($D^\alpha T$, $fT$, $\tau_h T$) suggests we are acting on $T$, but the operation is transferred to $\varphi$ via the adjoint. This is the adjoint construction from the duality chapter (Definition 1) applied concretely: if $A : \mathcal{D}(\Omega) \to \mathcal{D}(\Omega)$ is a continuous linear map, its adjoint $A^* : \mathcal{D}'(\Omega) \to \mathcal{D}'(\Omega)$ acts by $A^* T = T \circ A$. Differentiation, multiplication, and translation on distributions are all instances of this. The notation is justified by consistency: when $T = T_f$ for a classical function $f$, the distribution $AT$ agrees with the distribution $T_{Af}$ defined by applying $A$ to $f$ directly.

Multiplication by Smooth Functions¶

Given $f \in C^\infty(\Omega)$ and $T \in \mathcal{D}'(\Omega)$, we want to define a new distribution that deserves the name “$fT$.”

Note that $U$ is well-defined as a map: $f\varphi \in \mathcal{D}(\Omega)$ whenever $\varphi \in \mathcal{D}(\Omega)$, because the product of a smooth function and a compactly supported smooth function is again compactly supported and smooth. So $T(f\varphi)$ makes sense.

Differentiation of Distributions¶

With the duality principle now familiar from translation and multiplication, we come to the main payoff: differentiation.

The definition¶

The definition is motivated by integration by parts. Suppose $f$ is smooth. Then for every $\varphi \in \mathcal{D}(\Omega)$,

since the boundary terms vanish (test functions have compact support in $\Omega$). In the language of distributions: $T_{f'}(\varphi) = -T_f(\varphi')$. The right-hand side makes sense for any distribution $T$, not just those coming from smooth functions: given $T \in \mathcal{D}'(\Omega)$, the map $\varphi \mapsto -T(\varphi')$ is perfectly well-defined. This suggests a definition.

This is the main payoff of the distributional framework: differentiation is always possible, and it always produces another distribution.

The differentiation cascade¶

The power of distributional derivatives is best seen through a chain of examples in one dimension ($\Omega = \mathbb{R}$).

Properties of distributional differentiation¶

Property 4 is remarkable: you can always interchange limits and derivatives in the sense of distributions. In classical analysis, this requires uniform convergence of derivatives. In the distributional framework, pointwise convergence of the functionals is enough, because the derivative is transferred to the (fixed, smooth) test function.

Taylor’s theorem for distributions¶

Now that we have both translation and differentiation, we can combine them into a distributional Taylor expansion. The idea is simple: to expand $\tau_h T$ in powers of $h$, apply $\tau_h T$ to a test function and use the classical Taylor expansion of $\varphi(x + h)$ in $h$.

Weak Derivatives¶

Distributional derivatives exist for every distribution, but they may not be representable by a function. When they are, we get the notion of a weak derivative, which bridges distribution theory and Sobolev spaces.

In the language of adjoints: the distributional derivative $D^\alpha T_u$ is the functional $\varphi \mapsto (-1)^{|\alpha|} \int u\, D^\alpha\varphi\, dx$. This always defines an element of $\mathcal{D}'(\Omega)$. The question is whether this functional can be represented by a function $v$ via $\varphi \mapsto \int v\,\varphi\, dx$. When it can, $v = D^\alpha u$ is the weak derivative of $u$. If moreover $u \in L^p$ and $D^\alpha u \in L^p$, then $u$ belongs to a Sobolev space $W^{k,p}(\Omega)$: the space of $L^p$ functions whose weak derivatives up to order $k$ are also in $L^p$. Sobolev spaces are the natural home for PDE solutions that are not classically differentiable, and their theory rests entirely on the distinction between distributional and weak derivatives.

Translation and Reflection¶

The simplest operations on distributions illustrate the duality principle with no calculus required.

Recall that for functions on $\mathbb{R}^d$, translation by $h \in \mathbb{R}^d$ is $(\tau_h \varphi)(x) = \varphi(x - h)$ and reflection is $\check{\varphi}(x) = \varphi(-x)$. Given a distribution $T \in \mathcal{D}'(\mathbb{R}^d)$, we want to define the translated distribution $\tau_h T$ and the reflected distribution $\check{T}$.

Convolution¶

Convolution with a test function is a smoothing operation: it always produces a smooth function, even when applied to a distribution. The definition uses translation and reflection from the preceding section.

Convolution of functions (review)¶

For $f \in L^1_{\mathrm{loc}}(\mathbb{R}^d)$ and $\psi \in \mathcal{D}(\mathbb{R}^d)$:

The result $f * \psi$ is smooth ($C^\infty$) and satisfies $D^\alpha(f * \psi) = f * D^\alpha \psi$.

Convolution of a distribution with a test function¶

When $T = T_f$ for a locally integrable function $f$, this recovers the usual convolution: $(T_f * \psi)(x) = \int f(y)\,\psi(x-y)\,dy = (f * \psi)(x)$.

Approximation to the identity¶

This is one of the most useful results in analysis: every distribution can be approximated by smooth functions. This is the distributional analogue of sequential density (Proposition 3), made constructive.