In this chapter we study a class of distributions, summable distributions, that can be extended on smooth bounded functions. These distributions have properties that are well suited to describe some classes of systems. However, the material is more technical than the rest of the book and can be skipped without loss of continuity.

1 Definition and Canonical Extension

One can define summable distributions in several equivalent ways [16]. The following one is the most suitable for our purposes.

Definition 6.1

(Summable distributions) A summable distribution T is a distribution that can be represented as a finite sum of derivatives (in the sense of distribution) of functions \(f_k\in L^1\)

$$\begin{aligned} T = \sum _{|k|\le m} D^kf_k \end{aligned}$$

with k an n-tuple in \({\mathbb {N}}^n\) and \(m\in {\mathbb {N}}\).

We denote the vector space of summable distributions by  \({\mathcal {D}}_{L^1}'\).

An important property of summable distributions is the fact that they can be extended to continuous linear functionals on  \({\mathcal {B}}\), the set of indefinitely differentiable functions that, together with all their derivatives, are  bounded

$$\begin{aligned} {\mathcal {B}} :=\left\{ \phi \in {\mathcal {C}}^\infty |\, D^k\phi \ \text {is bounded},\ k\in {\mathbb {N}}^n \right\} . \end{aligned}$$

As usual, to talk about continuity, we need to define a convergence criterion (topology).

Definition 6.2

(Convergence in \(\boldsymbol{{\mathcal {B}}}\)) A sequence of functions \((\eta _j)\) in \({\mathcal {B}}\) converges to zero if the sequence as well as all sequences of the derivatives \((D^k\eta _j)\) converge uniformly to zero as j tends to infinity. That is, given the norms

$$\begin{aligned} p_m(\eta ) :=\sum _{|k|\le m} \sup _{\tau \in {\mathbb {R}}^n}|D^k\eta (\tau )|, \end{aligned}$$

the sequence \((\eta _j)\) converges to zero if, as j tends to infinity, the sequence of numbers \((p_m(\eta _j))\) converges to zero for all \(m\in {\mathbb {N}}\).

The application of a summable distribution T to a function \(\eta \in {\mathcal {B}}\) is well defined since

$$\begin{aligned} \left| \left\langle D^kf_k,\eta \right\rangle \right| \le \sup _{\tau \in {\mathbb {R}}^n}|D^k\eta (\tau )| \int _{{\mathbb {R}}^n}\,\left| f_k(\tau )\,\right| \,d^n\tau < \infty \,. \end{aligned}$$

This shows that it’s possible to extend a summable distribution to \({\mathcal {B}}\). However, the extension in general is not unique. The reason being that the set of test functions \({\mathcal {D}}\) is not dense in \({\mathcal {B}}\). That is to say that it’s not possible to approximate to arbitrary accuracy any function in \({\mathcal {B}}\) with functions from \({\mathcal {D}}\).

Example 6.1: Constant Function

Consider the constant function \(\textbf{1}: t\mapsto 1\), the function \(\alpha \in {\mathcal {D}}\) defined by (2.11), and functions \(\alpha _j\in {\mathcal {D}}\) defined by \(\alpha _j(t) = \alpha (t/j)\), \(j\in {\mathbb {N}}\). The product \(\alpha _j\textbf{1}\) is clearly a test function satisfying \(\alpha _j(2j) = 0\) for all values of j. From this we see that

$$\begin{aligned} p_0(\alpha _j\textbf{1} - \textbf{1}) = \sup _{t\in {\mathbb {R}}}|\alpha _j(t)\textbf{1}(t) - \textbf{1}(t)| = 1 \end{aligned}$$

no matter how large j is.

Example 6.2: Average Functional

Consider the functions \(\textbf{1},\alpha _j\) from Example 6.1 and the following functional L on \({\mathcal {B}}\)

$$\begin{aligned} L(\eta ) :=\lim _{C\rightarrow \infty }\frac{1}{C}\int _{-C/2}^{C/2}\eta (\tau )\,d\tau \,. \end{aligned}$$

It is easily seen that L is linear and continuous. Its value on the constant function \(\textbf{1}\) is 1 while its value on the test functions \(\alpha _j\textbf{1}\) is zero for all values of j

$$\begin{aligned} |L(\alpha _j\textbf{1})| \le \lim _{C\rightarrow \infty }\frac{1}{C}\int _{-2j}^{2j}|\alpha _j(\tau )|\,d\tau \le \lim _{C\rightarrow \infty }\frac{4j}{C} = 0 . \end{aligned}$$

The functional L is therefore a valid extension to \({\mathcal {B}}\) of the zero distribution as is the zero functional on \({\mathcal {B}}\).

We can define a unique, canonical extension of a summable distribution T by requiring an additional condition on the extension [19]. A suitable condition can be obtained from the properties of Lebesgue integrals. Consider again the test functions \(\alpha _j\in {\mathcal {D}}\) from Example 6.1 and a function \(\eta \in {\mathcal {B}}\). If we apply T to \(\alpha _j\eta \) we obtain

$$\begin{aligned} &\,\, \left\langle T,\alpha _j\eta \right\rangle = \sum _{|k|\le m} (-1)^{|k|}\int _{{\mathbb {R}}^n}\, f_k(\tau )\,D^k(\alpha _j(\tau )\eta (\tau ))\,d^n\tau \\ & = \sum _{|k|\le m} (-1)^{|k|}\left( \int _{|\tau |\le j} f_k(\tau )\,D^k\eta (\tau )\,d^n\tau + \int _{|\tau |>j} f_k(\tau )\,D^k(\alpha _j(\tau )\eta (\tau ))\,d^n\tau \right) \,. \end{aligned}$$

The integral of an \(L^1\) function can be approximated up to an arbitrary \(\epsilon > 0\) by an integral over a suitably chosen compact subset K of \({\mathbb {R}}^n\). Therefore we can find a large enough N such that for \(j>N\)

$$\begin{aligned} \left\langle T,\alpha _j\eta \right\rangle = \epsilon + \sum _{|k|\le m}(-1)^{|k|} \int _{|\tau |\le j}f_k(\tau )D^k\eta (\tau )\,d^n\tau \,. \end{aligned}$$

Thus, in the limit as j tends to infinity we obtain a well-defined continuous linear functional on \({\mathcal {B}}\).

The important observation from this derivation is the fact that, to find an extension to \({\mathcal {B}}\) of a summable distribution, it is not necessary to require uniform convergence on the whole of \({\mathbb {R}}^n\). An extension can be obtained by requiring uniform convergence on every compact subset \(K\subset {\mathbb {R}}^n\). More precisely, by requiring the convergence criterion that we defined for the space \({\mathcal {E}}\). From this observation we define the following property.

Definition 6.3

(Bounded convergence property) A continuous linear functional on \({\mathcal {B}}\) has the bounded convergence property if, given any sequence \((\eta _j)\) of functions \(\eta _j\in {\mathcal {B}}\) with \(p_m(\eta _j)<\infty \) for all m, and converging to zero in the space \({\mathcal {E}}\) as \(j\rightarrow \infty \), then

$$\begin{aligned} \left\langle T,\eta _j \right\rangle \rightarrow 0\,, \qquad j\rightarrow \infty \,. \end{aligned}$$

The sequence \((\alpha _j\eta )\) does converge to \(\eta \) in \({\mathcal {E}}\). Hence, by continuity, there is a unique extension of T to \({\mathcal {B}}\) with the bounded convergence property

$$\begin{aligned} \lim _{j\rightarrow \infty }\left\langle T,\alpha _j\eta \right\rangle = \left\langle T,\eta \right\rangle \,. \end{aligned}$$
(6.1)

In particular this shows that this extension does not depend on the particular representation of T in terms of derivatives of integrable functions.

The converse is also true. The restriction to \({\mathcal {D}}\) of any continuous linear functional on \({\mathcal {B}}\) with the bounded convergence property defines a unique summable distribution. Thus, there is a one to one correspondence between summable distributions and continuous linear functionals on \({\mathcal {B}}\) with the bounded convergence property.

Definition 6.4

(Canonical extension) The canonical extension to \({\mathcal {B}}\) of a summable distribution is the unique extension to a continuous linear functional on \({\mathcal {B}}\) with the bounded convergence property.

In the following, whenever we use the extension of a summable distribution it will always be assumed to be the canonical one.

While our previous definition of differentiation carries over to summable distributions without problems, this is not the case for multiplication. In general the product of a bounded function \(\eta \in {\mathcal {B}}\) with an unbounded one \(\gamma \in {\mathcal {E}}\) is not bounded and therefore not in \({\mathcal {B}}\). Differently from this, the product of two bounded functions \(\eta ,\zeta \in {\mathcal {B}}\) is always in \({\mathcal {B}}\). Therefore, for summable distributions \(T \in {\mathcal {D}}_{L^1}'\) multiplication has to be restricted to functions in \({\mathcal {B}}\)

$$\begin{aligned} \left\langle \eta T,\zeta \right\rangle = \left\langle T,\eta \zeta \right\rangle \,. \end{aligned}$$

2 Convolution of Summable Distributions

In Sect. 3.2 we defined the convolution product between two distributions \(S,T \in {\mathcal {D}}'\) by

$$\begin{aligned} \left\langle S *T,\phi \right\rangle = \left\langle S(\tau ) \otimes T(\lambda ),\phi (\tau + \lambda ) \right\rangle \end{aligned}$$

and saw that in general, if the support of both S and T is unbounded, it may not exist. In this section we show that if S and T are summable, then their convolution product is well-defined despite the fact that their support is unbounded.

Consider the application of a summable distribution T to a function \(\tau \mapsto \eta (\lambda + \tau ) \in {\mathcal {B}}\) with \(\lambda \) a parameter. Following the same arguments as in Sect. 3.1, given the linearity and continuity of T, we deduce that it is a continuous and indefinitely differentiable function \(\zeta \) belonging to \({\mathcal {B}}\)

$$\begin{aligned} \zeta (\lambda ) = \left\langle T(\tau ),\eta (\lambda + \tau ) \right\rangle \,. \end{aligned}$$

For this reason the convolution of two summable distributions S and T is always well-defined

$$\begin{aligned} \left\langle S *T,\eta \right\rangle = \left\langle S(\lambda ),\left\langle T(\tau ),\eta (\lambda + \tau ) \right\rangle \right\rangle = \left\langle S,\zeta \right\rangle \end{aligned}$$

and commutative.

Next we investigate the convolution of a summable distribution T with a function in \({\mathcal {B}}\). Consider the application of T to the function \(\tau \mapsto \eta (\lambda - \tau ) \in {\mathcal {B}}\) parameterised by \(\lambda \). As we just saw, it is a function that we call again \(\zeta \) and that is clearly locally integrable. Hence, it defines a distribution in \({\mathcal {D}}'\) and with \(\phi \in {\mathcal {D}}\) we can write

$$\begin{aligned} \begin{aligned} \left\langle \left\langle T(\tau ),\eta (\lambda - \tau ) \right\rangle ,\phi (\lambda ) \right\rangle &= \left\langle \zeta ,\phi \right\rangle = \left\langle \phi ,\zeta \right\rangle = \left\langle \phi (\lambda ),\left\langle T(\tau ),\eta (\lambda - \tau ) \right\rangle \right\rangle \\ &= \left\langle \phi (\lambda ) \otimes T(\tau ),\eta (\lambda - \tau ) \right\rangle \\ &= \left\langle T(\tau ),\int _{{\mathbb {R}}^n} \phi (\lambda ) \eta (\lambda - \tau )\,d^n\lambda \right\rangle \\ &= \left\langle T(\tau ),\int _{{\mathbb {R}}^n} \phi (\xi + \tau ) \eta (\xi )\,d^n\xi \right\rangle \\ &= \left\langle T(\tau ) \otimes \eta (\xi ),\phi (\xi + \tau ) \right\rangle = \left\langle T *\eta ,\phi \right\rangle \,. \end{aligned} \end{aligned}$$

or

$$\begin{aligned} \left\langle T(\tau ),\eta (\lambda - \tau ) \right\rangle = (T *\eta )(\lambda ) \end{aligned}$$
(6.2)

This shows that a summable distribution can be regularised by a function in \({\mathcal {B}}\) and that the resulting regularised is also a function in \({\mathcal {B}}\).

3 Fourier Transform of Summable Distributions

The functions \(\tau \mapsto \textrm{e}^{-\jmath \left( \omega ,\tau \right) }\) with \(\omega \in {\mathbb {R}}^n\) belong to \({\mathcal {B}}\). For this reason the Fourier transform of a summable distribution T can be expressed in a simple way. Let \(\phi \in {\mathcal {D}}\), then

$$\begin{aligned} \begin{aligned} \left\langle {\mathcal {F}}\{T\},\phi \right\rangle &= \left\langle T(\tau ),{\mathcal {F}}\{\phi \}(\tau ) \right\rangle = \left\langle {{T(\tau )}}, \int _{{\mathbb {R}}^n}\phi (\omega )\textrm{e}^{-\jmath \left( \tau ,\omega \right) }\,d^n\omega \right\rangle \\ &= \left\langle T(\tau ),\left\langle \phi (\omega ),\textrm{e}^{-\jmath \left( \tau ,\omega \right) } \right\rangle \right\rangle = \left\langle T(\tau )\otimes \phi (\omega ),\textrm{e}^{-\jmath \left( \tau ,\omega \right) } \right\rangle \\ &= \left\langle \phi (\omega ),\left\langle T(\tau ),\textrm{e}^{-\jmath \left( \tau ,\omega \right) } \right\rangle \right\rangle = \left\langle \left\langle T(\tau ),\textrm{e}^{-\jmath \left( \tau ,\omega \right) } \right\rangle ,\phi (\omega ) \right\rangle \end{aligned} \end{aligned}$$

or

$$\begin{aligned} {\mathcal {F}}\{T\}(\omega ) = \left\langle T(\tau ),\textrm{e}^{-\jmath \left( \tau ,\omega \right) } \right\rangle \,. \end{aligned}$$
(6.3)

\({\mathcal {F}}\{T\}\) is thus a continuous function. Moreover it has at most polynomial growth, for, by representing T as a sum of integrable functions and the properties of the Fourier transform, for some \(m \in {\mathbb {N}}\) we have

$$\begin{aligned} \begin{aligned} \left|{\mathcal {F}}\{T\}(\omega ) \right| &= \left|\sum _{|k|\le m}(\jmath \omega )^k{\mathcal {F}}\{f_k\}(\omega ) \right|\\ &\le \sum _{|k|\le m}\left|\omega \right|^k\int _{{\mathbb {R}}^n}\left|f_k(\tau ) \right|\,d^n\tau \le C (1 + \left|\omega \right|)^m \end{aligned} \end{aligned}$$

with C a constant. Thus, the Fourier transformed of a summable distribution is a function of slow growth.

The converse is not in general true, but we can find a class of functions for which it is. This is the set  \({\mathcal {O}}_M\), the set of functions of slow growth that are indefinitely differentiable.

To see that this is the case, consider the Fourier transformed \(\hat{T}\) of some tempered distribution T and assume that \(\hat{T} \in {\mathcal {O}}_M\). If \(\phi \in {\mathcal {D}} \subset {\mathcal {S}}\) then its Fourier transformed \(\hat{\phi }\) as well as \(\hat{\phi }\hat{T}\) are in \({\mathcal {S}}\). Therefore we see that

$$\begin{aligned} \phi *T = {\mathcal {F}}^{-1}\{\hat{\phi }\hat{T}\} \in {\mathcal {S}} \subset L^1 \end{aligned}$$

is a summable distribution and we can apply it to a function \(\eta \in {\mathcal {B}}\) to obtain

$$\begin{aligned} \left\langle \phi *T,\eta \right\rangle = \left\langle \phi (\lambda ) \otimes T(\tau ),\eta (\lambda + \tau ) \right\rangle = \left\langle T(\tau ),\left\langle \phi (\lambda ),\eta (\lambda + \tau ) \right\rangle \right\rangle \,. \end{aligned}$$

Since \(\phi \in {\mathcal {D}}\) and \(\eta \in {\mathcal {B}}\) are arbitrary and \(\left\langle \phi (\lambda ),\eta (\lambda + \tau ) \right\rangle \in {\mathcal {B}}\) we deduce that T is a summable distribution.

We conclude this section by showing that the property of the Fourier transform of transforming convolution products into ordinary products is valid for arbitrary summable distributions. Let S, T be summable distributions, then using (6.3) and the property of the exponential function \(\textrm{e}^{-\jmath \left( \tau +\lambda ,\omega \right) } = \textrm{e}^{-\jmath \left( \tau ,\omega \right) }\textrm{e}^{-\jmath \left( \lambda ,\omega \right) }\) one readily obtain that

$$\begin{aligned} {\mathcal {F}}\{S *T\} = {\mathcal {F}}\{S\} {\mathcal {F}}\{T\}\,. \end{aligned}$$
(6.4)

The product is well defined as \({\mathcal {F}}\{S\}\) and \({\mathcal {F}}\{T\}\) are both functions.