Convolution of Distributions

Abstract

The convolution product plays a central role in the description of linear and weakly-nonlinear systems. In this chapter we develop its theory based on distributions. In addition, the chapter introduces the tensor product which will also come to play an important role in the description of weakly-nonlinear systems.

The convolution product plays a central role in the description of linear and weakly-nonlinear systems. In this chapter we develop its theory based on distributions. In addition, the chapter introduces the tensor product which will also come to play an important role in the description of weakly-nonlinear systems.

1 Tensor Product

The general notion of convolution of distributions is defined in terms of the tensor product. We therefore start by defining this product and, as before, we start by considering regular distributions.

The tensor product is a bilinear operation that can be used to generate a vector space out of other vector spaces. If f is a function on ${\mathbb {R}}^m$ and g a function on ${\mathbb {R}}^n$, then the tensor product of f and g is defined as the function

$$\begin{aligned} f \otimes g : {\mathbb {R}}^{m+n} \rightarrow {\mathbb {C}}\quad (\tau ,\lambda ) \mapsto f(\tau )g(\lambda ) \,. \end{aligned}$$

The tensor product of two locally integrable functions is itself locally integrable. Therefore, if we now assume f and g to be locally integrable, we can try to build the tensor product of the regular distributions $T_f$ and $T_g$ based on the tensor product $f \otimes g$. If we use as test function the tensor product of two suitable test functions $\xi$ and $\psi$ then we obtain

$$\begin{aligned} \langle T_f \otimes T_g,\xi \otimes \psi \rangle = \langle T_f,\xi \rangle \langle T_g,\psi \rangle \end{aligned}$$

which is well-defined. However, for an arbitrary test function $\phi \in {\mathcal {D}}({\mathbb {R}}^{m+n})$ it is not immediately apparent that the result is a distribution. Taking $m=n=1$ for simplicity, we have

$$\begin{aligned} \langle T_f \otimes T_g,\phi \rangle = & {} \int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty \, f(\tau ) \, g(\lambda ) \, \phi (\tau ,\lambda ) \, d\lambda \, d\tau \nonumber \\ = & {} \int \limits _{-\infty }^\infty \, f(\tau ) \, \int \limits _{-\infty }^\infty \, g(\lambda ) \, \phi (\tau ,\lambda ) \,d\lambda \, d\tau \,. \end{aligned}$$

The inner integral evaluates to a number for every value of the variable $\tau$, that is, it is a complex valued function of $\tau$ that we call $\zeta (\tau )$. Furthermore, the variable $\tau$ only appears as an argument of the test function $\phi$. Therefore $\zeta$ must have compact support. In addition, when computing its derivative, differentiation can be moved under the integral and $\zeta$ is therefore indefinitely differentiable. In other words $\zeta$ is a test function. We therefore have

$$\begin{aligned} \langle T_f \otimes T_g,\phi \rangle = \langle T_f,\langle T_g,\phi \rangle \rangle = \langle T_g,\langle T_f,\phi \rangle \rangle \end{aligned}$$

where the last equality comes from the fact that we could reverse the order of integration without changing the result. This last property is referred to as  Fubini’s theorem.

The above arguments can be generalized to arbitrary distributions. That the inner functional is a function of $\tau$ and that has compact support is clear. The fact that it can be differentiated comes from the continuity and linearity of distributions

$$\begin{aligned} D\,\zeta (\tau ) = & {} \lim _{\epsilon \rightarrow 0} \frac{\zeta (\tau + \epsilon ) - \zeta (\tau )}{\epsilon } \nonumber \\ = & {} \lim _{\epsilon \rightarrow 0} \frac{\langle T(\lambda ),\phi (\tau + \epsilon , \lambda )) \rangle - \langle T(\lambda ),\phi (\tau , \lambda ) \rangle }{\epsilon } \nonumber \\ = & {} \lim _{\epsilon \rightarrow 0} \langle T(\lambda ),\frac{\phi (\tau + \epsilon , \lambda ) - \phi (\tau , \lambda )}{\epsilon } \rangle \nonumber \\ = & {} \langle T(\lambda ),D_1 \phi (\tau , \lambda ) \rangle \,. \end{aligned}$$

(3.1)

With this we see that $\zeta$ can be differentiated an arbitrary number of times and is thus a test function. We therefore obtain the following general definition for the tensor product of distributions.

Definition 3.1

(Tensor product)  Given two distributions $S \in {\mathcal {D}}'({\mathbb {R}}^m)$ and $T \in {\mathcal {D}}'({\mathbb {R}}^n)$ the tensor product $S \otimes T$ is the distribution in ${\mathcal {D}}'({\mathbb {R}}^{m+n})$ defined by

$$\begin{aligned} \langle S \otimes T,\phi \rangle :=\langle S,\langle T,\phi \rangle \rangle = \langle T,\langle S,\phi \rangle \rangle \,. \end{aligned}$$

(3.2)

It’s easy to see that the tensor product of distributions is bilinear

$$\begin{aligned} \begin{aligned} (S + T) \otimes U &= S \otimes U + T \otimes U \\ S \otimes (T + U) &= S \otimes T + S \otimes U\,, \end{aligned} \end{aligned}$$

(3.3)

and associative

$$\begin{aligned} (S \otimes T) \otimes U = S \otimes (T \otimes U)\,. \end{aligned}$$

(3.4)

As a useful abbreviation of notation we define the tensor power by

$$\begin{aligned} \begin{aligned} T^{\otimes k} &:=\underbrace{T \otimes \ldots \otimes T}_{k\ \text {times}}\,, \qquad k > 0\\ T^{\otimes 0} &:=1 \in {\mathbb {C}}\,. \end{aligned} \end{aligned}$$

(3.5)

Example 3.1: Higher Dimensional Dirac Pulse

The tensor product of two Dirac pulses is

$$\begin{aligned} \langle \delta \otimes \delta (\tau , \lambda ),\phi (\tau ,\lambda ) \rangle = & {} \langle \delta (\tau ),\langle \delta (\lambda ),\phi (\tau ,\lambda ) \rangle \rangle = \langle \delta (\tau ),\phi (\tau ,0) \rangle \\ = & {} \phi (0,0) \\ =: & {} \langle \delta (\tau ,\lambda ),\phi (\tau ,\lambda ) \rangle \,. \end{aligned}$$

2 Convolution of Distributions

We now come to the main objective of this section: the convolution of distributions. Remember that the convolution of integrable functions $f,g \in L^1$ is defined as follows

$$\begin{aligned} f *g \, (t) :=\int \limits _{-\infty }^\infty \,f(\tau )\,g(t-\tau )\,d\tau \,. \end{aligned}$$

To obtain a distribution we may write

$$\begin{aligned} \langle f *g,\phi \rangle = & {} \int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty \, f(\tau )\,g(t-\tau ) \, d\tau \, \phi (t) \, dt \\ = & {} \int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty \, f(\tau )\,g(\lambda ) \, \phi (\lambda + \tau ) \, d\tau \, d\lambda \end{aligned}$$

which can be represented as the following tensor product

$$\begin{aligned} \langle f *g,\phi \rangle = \langle f(\tau ) \otimes g(\lambda ),\phi (\lambda + \tau ) \rangle \,. \end{aligned}$$

However, while indefinitely differentiable, the function $\psi (\tau ,\lambda ) = \phi (\lambda + \tau )$ is not a test function because its support is not compact. In fact, $\psi (\tau ,\lambda )$ assumes the same value $\phi (t)$ for every point on the diagonal line $t = \lambda + \tau$ of the $(\tau ,\lambda )$-plane (see Fig. 3.1). In spite of this, on account of our assumption that f and g are integrable functions (and not merely locally integrable), the above integral is well-defined. We therefore conclude that, similarly to the case of functions, the convolution of distributions only exists for a subset of distributions with additional characteristics.

Fig. 3.1

Support of $\psi (\tau ,\lambda ) = \phi (\tau + \lambda )$

Definition 3.2

(Convolution)  Given two distributions S and T in ${\mathcal {D}}'({\mathbb {R}}^n)$, if for every test function $\phi \in {\mathcal {D}}({\mathbb {R}}^n)$ the tensor product $S \otimes T$ can be extended to functions of the form $\psi (\tau ,\lambda ) = \phi (\tau + \lambda )$, then the convolution product $S *T$ is defined by

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

(3.6)

and is commutative

$$\begin{aligned} S *T = T *S\,. \end{aligned}$$

(3.7)

A sufficient condition for the existence of the convolution is as follows: if the intersection of the support of $S \otimes T$, that is $\text {supp}(S \otimes T) = \text {supp}(S) \times \text {supp}(T)$ and the support of $\psi (\tau ,\lambda ) = \phi (\tau +\lambda )$ is bounded, then $S *T$ is well defined. In other words, if for $\tau \in \text {supp}(S)$ and $\lambda \in \text {supp}(T)$ the sum $\tau + \lambda$ can only remain bounded if both $\tau$ and $\lambda$ remain bounded, then the convolution product $S *T$ is well defined.

Note that this condition is sufficient but not necessary as shown for instance by the introductory example with integrable functions $f,g \in L^1$. In fact the convolution $f *g$ of integrable functions does always exist and is itself an integrable function

$$\begin{aligned} \left|\langle f *g,\phi \rangle \right| {} & {} = \left|\int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty f(\tau )\,g(\lambda ) \, \phi (\lambda + \tau ) \, d\tau \, d\lambda \right| \\ {} & {} \le \sup {|\phi |} \, \int \limits _{-\infty }^\infty |f(\tau )| \, d\tau \, \int \limits _{-\infty }^\infty |g(\lambda )| \, d\lambda \,. \end{aligned}$$

Fig. 3.2

Support of $S(\tau )\otimes T(\lambda )$ and of $\psi (\tau ,\lambda ) = \phi (\tau + \lambda )$

Example 3.2: One Sided Distributions

A subset of the real line $U \in {\mathbb {R}}$ is said to be bounded on the left if there is a real constant b such that $U \subset (b, \infty )$. Similarly, a subset U is called bounded on the right if there is a constant b such that $U \subset (-\infty , b)$.

Distributions whose support is bounded on the left (right) are called right-sided  (left-sided) distributions. The set of all such distributions forms a vector space denoted by ${\mathcal {D'}}_R$ ( ${\mathcal {D'}}_L$). Of particular interest for our purposes are right-sided distributions T with $\text {supp}(T) \in [0, \infty )$. We denote the space of all such distributions by ${\mathcal {D_+'}}$.

Figure 3.2 shows the support of $S(\tau ) \otimes T(\lambda )$ and of $\psi (\tau ,\lambda ) = \phi (\tau + \lambda )$ for two distributions S and T in ${\mathcal {D_+'}}$. It is clear that, for any test function $\phi$, their overlap is always bounded. Therefore, the convolution of right-sided or left-sided distributions is always well defined. Not so the convolution of a left-sided distribution with a right-sided one.

Example 3.3: Convolution with $\boldsymbol{\delta }$

Let T be any distribution in ${\mathcal {D}}'({\mathbb {R}}^n)$ and $\delta$ the n dimensional Dirac pulse (see Example 3.1), then

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

or

$$\begin{aligned} T *\delta = T \,. \end{aligned}$$

(3.8)

Thus, $\delta$ is a unit of convolution.

Similarly, for any |k|th order derivative of the Dirac pulse

$$\begin{aligned} \langle T *D^k\delta ,\phi \rangle = & {} \langle T(\tau ) \otimes D^k\delta (\lambda ),\phi (\tau + \lambda ) \rangle \\ = & {} \langle T(\tau ), \langle D^k\delta (\lambda ),\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle T(\tau ), \langle \delta (\lambda ),(-1)^{|k|}D_\lambda ^k\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle T(\tau ), \langle \delta (\lambda ),(-1)^{|k|}D_\tau ^k\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle T(\tau ),(-1)^{|k|}D^k\phi (\tau ) \rangle \\ = & {} \langle D^kT(\tau ),\phi (\tau ) \rangle \end{aligned}$$

or

$$\begin{aligned} T *D^k\delta = D^kT \end{aligned}$$

(3.9)

where $D_\lambda ^k$ and $D_\tau ^k$ mean differentiation with respect to the variable $\lambda$ and $\tau$, respectively; and we made use of the fact that $D_\lambda ^k\phi (\tau + \lambda ) = D_\tau ^k\phi (\tau + \lambda )$.

In our notation we use T(t) to indicate a distribution to be associated with a test function whose independent variable is indicated by the symbol t. With this notation it seems natural to write $S(t) *T(t)$ to denote the convolution of two distributions. However, when we build the convolution of two shifted distributions this leads to confusion as $S(t - a) *T(t - a)$ does not represent the distribution $S *T$ shifted by a. To give a precise meaning to such expressions we introduce the shifting operator defined by

$$\begin{aligned} \langle \tau _{a} T,\phi (t) \rangle :=\langle T(t),\phi (t + a) \rangle \,. \end{aligned}$$

(3.10)

With it we fix the following notation

$$\begin{aligned} (S *T)(t - a) := & {} \tau _{a}(S *T) \end{aligned}$$

(3.11)

$$\begin{aligned} S(t - a) *T(t - b) := & {} \tau _{a}S *\tau _{b}T\,. \end{aligned}$$

(3.12)

The convolution product has several useful properties. The first one that we want to discuss is distributivity. If all appearing convolutions are well defined, then

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

(3.13)

and similarly

$$\begin{aligned} S *(T + U) = & {} S *T + S *U\,. \end{aligned}$$

(3.14)

Further, differentiation of a convolution product is equivalent to differentiation of one of the products

$$\begin{aligned} \langle D_i(S *T),\phi \rangle = & {} -\langle S *T,D_i\phi \rangle \\ = & {} -\langle S(\tau ) \otimes T(\lambda ),D_{\tau ,i}\phi (\tau + \lambda ) \rangle \\ = & {} -\langle S(\tau ),\langle T(\lambda ),D_{\tau ,i}\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle D_i S(\tau ),\langle T(\lambda ),\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle (D_i S) *T,\phi \rangle \end{aligned}$$

where $D_{\tau ,i}$ is the partial differential operator with respect to the ith component of the variable $\tau$. Since $D_{\tau ,i}\phi (\tau + \lambda ) = D_{\lambda ,i}\phi (\tau + \lambda )$ differentiation can also be moved to the second factor so that

$$\begin{aligned} D_i(S *T) = (D_i S) *T = S *(D_i T)\,. \end{aligned}$$

(3.15)

In a similar way one shows that the operation of shifting a convolution product can also be moved to one of the factors

$$\begin{aligned} (S *T)(\tau - a) = S(\tau - a) *T(\tau ) = S(\tau ) *T(\tau - a)\,. \end{aligned}$$

(3.16)

Example 3.4: Convolution with $\boldsymbol{\delta }$

Consider two Dirac pulses and an arbitrary distribution T in ${\mathcal {D}}'({\mathbb {R}}^n)$. By the shifting property of convolution we have

$$\begin{aligned} \delta (\tau - a) *\delta (\tau - b) = & {} \delta (\tau - a - b) \end{aligned}$$

(3.17)

$$\begin{aligned} T(\tau ) *\delta (\tau - a) = & {} T(\tau - a) \,. \end{aligned}$$

(3.18)

The convolution of a distribution T with an indefinitely differentiable function $\gamma$ is an indefinitely differentiable function. For, by keeping in mind that $\phi$ has compact support and therefore, as a distribution can be uniquely extended to functions in ${\mathcal {E}}$, we have

$$\begin{aligned} \langle T *\gamma ,\phi \rangle = & {} \langle T(\tau ),\langle \gamma (\lambda ),\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle T(\tau ),\langle \gamma (\lambda - \tau ),\phi (\lambda ) \rangle \rangle \\ = & {} \langle T(\tau ),\langle \phi (\lambda ),\gamma (\lambda - \tau ) \rangle \rangle \\ = & {} \langle \phi (\lambda ),\langle T(\tau ),\gamma (\lambda - \tau ) \rangle \rangle \,. \end{aligned}$$

Then, by arguments similar to the ones that led to the definition of the tensor product, one deduces that the inner distribution is an indefinitely differentiable function that we call $\zeta$. We can therefore proceed further

$$\begin{aligned} \langle T *\gamma ,\phi \rangle = & {} \langle \phi (\lambda ),\zeta (\lambda ) \rangle \\ = & {} \langle \zeta ,\phi \rangle \end{aligned}$$

and obtain as claimed that $T *\gamma = \zeta$.

The convolution product is a continuous operation in the following sense. If T is a fixed convolution, $(S_m)_{m\in {\mathbb {N}}}$ a sequence of distributions converging in ${\mathcal {D'}}$ to S and all involved convolutions are well defined, then

$$\begin{aligned} \lim _{m\rightarrow \infty } \langle S_m *T,\phi \rangle = & {} \lim _{m\rightarrow \infty } \langle S_m(\tau ),\langle T(\lambda ),\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle S(\tau ),\langle T(\lambda ),\phi (\tau + \lambda ) \rangle \rangle \\ = & {} \langle S *T,\phi \rangle \end{aligned}$$

or

$$\begin{aligned} \lim _{m\rightarrow \infty } S_m *T = S *T\,. \end{aligned}$$

(3.19)

In particular we saw in Example 2.5 that $\delta$ can be represented as the limit of a sequence of test functions $\beta _m$ and in Example 3.3 that $\delta$ is a unit of convolution. With continuity of convolutions we therefore deduce that each distribution is the limit of a sequence of indefinitely differentiable functions of the form $T *\phi$ with $\phi$ a test function. We saw an instance of this in Example 2.2.

The last property that we want to discuss in this section is associativity. In general the convolution of three or more distributions is not associative as is easily verified with simple examples.

Example 3.5: Convolution may not be Associative

Let’s denote by 1 and 0 the constant functions evaluating to one and zero, respectively. Then

$$\begin{aligned} 1 *(\delta ^{(1)} *\textsf{1}_{+}) = & {} 1 *\delta = 1\\ (1 *\delta ^{(1)}) *\textsf{1}_{+}= & {} 0 *\textsf{1}_{+}= 0 \\ (\delta ^{(1)} *\textsf{1}_{+}) *1 = & {} \delta *1 = 1 \\ \delta ^{(1)} *(\textsf{1}_{+}*1) = & {} \text {undefined} \,. \end{aligned}$$

We can guarantee associativity by imposing a restriction similar to the one for the existence of the convolution of two distributions. Let’s write the convolution of three distributions in terms of the tensor product

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

(3.20)

If the intersection of the support of $S(\tau ) \otimes T(\lambda ) \otimes U(\kappa )$ and the support of $\phi (\tau + \lambda + \kappa )$ is bounded, then, by the properties of the tensor product, the convolution is guaranteed to be associative. It’s easily verified that the following is a sufficient condition: if all, but possibly one distribution have compact support, then the convolution product is associative.

Example 3.6: One-Sided Distributions

Consider three distributions $S(\tau )$, $T(\lambda )$ and $U(\kappa )$ in ${\mathcal {D_+'}}$. Then $\tau , \lambda$ and $\kappa$ are $\ge 0$. If the value of $\tau + \lambda + \kappa$ is bounded then there is a constant c for which $\tau + \lambda + \kappa < c$. It follows that $\tau$ is bounded by $\tau < c - (\lambda + \kappa )$ and similarly for the other variables. The convolution of distributions in ${\mathcal {D_+'}}$ is therefore always associative. This is also true for distributions in ${\mathcal {D'}}_R$ and ${\mathcal {D'}}_L$.

In addition, it’s easily seen that ${\mathcal {D_+'}}$ is closed under convolution. That is, a convolution between distributions in ${\mathcal {D_+'}}$ results in another distribution in ${\mathcal {D_+'}}$.

The discussed properties of the convolution product are summarized in Table 3.1.

Table 3.1 Properties of the convolution product

3 Approximation of Distributions

In this section we show how the convolution product can be used to obtain approximations of arbitrary distributions.

We saw that if T is a distribution in ${\mathcal {D}}'$ and $\phi$ is a test function in ${\mathcal {D}}$ then the convolution product $T *\phi$ is an indefinitely differentiable function. Its support is not necessarily bounded. However, let $\alpha$ be the test function defined by (2.11) and set $\alpha _m(\tau ) = \alpha (\tau /m)$. Then for every $m\in {\mathbb {N}}$ the product $\alpha _m\cdot (T *\phi )$ is an indefinitely differentiable function with compact support and hence a test function.

Let $(\beta _m)$ be a sequence of test functions converging to the $\delta$ distribution. Then, with the continuity of convolution, we see that in ${\mathcal {D}}'$

$$\begin{aligned} \lim _{m\rightarrow \infty } \alpha _m\cdot (T *\beta _m) = T\,. \end{aligned}$$

(3.21)

This shows that every distribution in ${\mathcal {D}}'$ is the limit of a sequence of test functions in ${\mathcal {D}}$. In other words,  ${\mathcal {D}}$ is a dense sub-vector space of ${\mathcal {D}}'$. Every distribution can thus be approximated to an arbitrary accuracy by a test function in ${\mathcal {D}}$.

Next we construct another dense sub-vector space of ${\mathcal {D}}'$. For simplicity we only treat the one dimensional case and for brevity we write $\kappa _m$ for $\alpha _m\cdot (T*\beta _m)$. As we just discussed $\kappa _m$ is a test function for every $m\in {\mathbb {N}}$. Let $\phi$ be another arbitrary test function. Then, for every m, we can find constants a and b such that the interval [a, b] includes both, the support of $\kappa _m$ as well as the one of $\phi$. If we construct the finite sum of $\delta$ distributions weighted by $k_m$

$$\begin{aligned} S_{n,m} = \frac{b-a}{n} \sum _{j=1}^n \kappa _m(a + j\frac{b-a}{n})\delta (t - a - j\frac{b-a}{n}) \end{aligned}$$

and apply it to $\phi$ we obtain

$$\begin{aligned} \langle S_{n,m},\phi \rangle = \frac{b-a}{n} \sum _{j=1}^n \kappa _m(a + j\frac{b-a}{n})\phi (a + j\frac{b-a}{n})\,. \end{aligned}$$

In the limit as n tends to infinity we obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\langle S_{n,m},\phi \rangle = \int \limits _a^b \kappa _m(\tau )\phi (\tau )\, d\tau \,. \end{aligned}$$

By the choice of the interval [a, b] we can extend it to the whole of ${\mathbb {R}}$ without changing the value of the integral. Hence, by letting m tend to infinity we finally obtain

$$\begin{aligned} \lim _{m\rightarrow \infty } \int \limits _a^b \kappa _m(\tau )\phi (\tau )\, d\tau = \lim _{m\rightarrow \infty }\langle \kappa _m,\phi \rangle = \langle T,\phi \rangle \,. \end{aligned}$$

 We thus see that every distribution $T\in {\mathcal {D}}'$ is the limit of a finite sum of weighted Dirac pulses $S_n:=S_{n,n}$. That is, finite sums of weighted $\delta$ distributions form a dense sub-vector space of ${\mathcal {D}}'$.

Note that a regular spacing between the $\delta$ distributions is not necessary and was chosen purely for convenience. In general any distribution can be approximated by a finite sum of the following form

$$\begin{aligned} T_n = \sum _{j=1}^n a_{n,j} \,\delta (t - \tau _{n,j}) \end{aligned}$$

(3.22)

with $a_{n,j}\in {\mathbb {C}}$ and $\tau _{n,j}\in {\mathbb {R}}$ .

4 Convolution of Periodic Distributions

In this section we investigate periodic distributions and their convolution. One way to define periodic distributions is to define them in a similar way as periodic functions.

Definition 3.3

(Periodic distribution I) A periodic distribution T is a distribution for which there exist a positive number ${\mathcal {T}}$ such that for all test functions $\phi$

$$\begin{aligned} \langle T(\tau ),\phi (\tau ) \rangle = \langle T(\tau + {\mathcal {T}}),\phi (\tau ) \rangle . \end{aligned}$$

(3.23)

The smallest such number ${\mathcal {T}}$ is called the  fundamental period of the distribution.

Periodic distributions have unbounded support. For this reason the convolution of two periodic distributions as defined by (3.6) does not exist. By exploiting their periodicity it is however possible to find an alternative definition for periodic distributions that allows for a well defined convolution product.

Consider a regular distribution arising from a ${\mathcal {T}}$-periodic function f. By exploiting its periodicity we find that

$$\begin{aligned} \langle f,\phi \rangle = & {} \int \limits _{-\infty }^\infty f(t) \, \phi (t) \, dt \\ = & {} \sum _{m=-\infty }^\infty \int \limits _{a+m{\mathcal {T}}}^{a+(m+1){\mathcal {T}}} f(t) \, \phi (t) \, dt \\ = & {} \int \limits _a^{a+{\mathcal {T}}} f(t) \sum _{m=-\infty }^\infty \phi (t - m{\mathcal {T}}) \, dt \\ = & {} \int \limits _a^{a+{\mathcal {T}}} f(t) \, \Phi (t) \, dt \end{aligned}$$

with a a constant,

$$\begin{aligned} \Phi (t) = \sum _{m=-\infty }^\infty \phi (t - m {\mathcal {T}}) \end{aligned}$$

(3.24)

and where the exchange of summation and integration is justified by the fact that for every value of t the sum is finite. The function $\Phi$ is ${\mathcal {T}}$-periodic and indefinitely differentiable.

By introducing the identity

$$\begin{aligned} f(t) \equiv f^\circ ([t]) \qquad (t \in {\mathbb {R}}) \end{aligned}$$

(3.25)

with $[t]$ the equivalence class of real numbers modulo ${\mathcal {T}}$, we effectively and uniquely define a function $f^\circ$. By writing $[t]$ as ${\mathcal {T}}/(2\pi )[\varphi ]$ and noting that $[\varphi ]$ is an equivalence class modulo $2\pi$, we can think of $f^\circ$ as a function defined on a circle of radius ${\mathcal {T}}/(2\pi )$ at the origin of a plane, with $[\varphi ]$ the polar angle. With this interpretation, the equivalence class $[t]$ is seen to represent the distance along the arc of the circle ${\mathbb {T}}$ from the reference $[0]$. In the following, to simplify notation, we are going to write a representative for an equivalence class.

Conversely, given a function $f^\circ$, the identity (3.25) uniquely defines a periodic function f (see Fig. 3.3). The last integral above is therefore identical to the integral of $f^\circ \, \Phi ^\circ$ on the circle ${\mathbb {T}}$

$$\begin{aligned} \int \limits _a^{a+{\mathcal {T}}} f(t) \, \Phi (t) \, dt = \int \limits _{\mathbb {T}}f^\circ (p) \, \Phi ^\circ (p) \, dp . \end{aligned}$$

We have thus obtained that to every regular periodic distribution f there corresponds a continuous linear functional $f^\circ$ on indefinitely differentiable functions $\Phi ^\circ$ on the circle ${\mathbb {T}}$. The set of all the latter functions is denoted by  ${\mathcal {D}}({\mathbb {T}})$. This space is isomorphic to the vector sub-space of ${\mathcal {E}}$ consisting of all indefinitely differentiable ${\mathcal {T}}$-periodic functions $\Phi$ and from which it inherits the following definition of convergence.

Definition 3.4

(Convergence in ${\mathcal {D}}({\mathbb {T}})$) A sequence of functions $\Phi _m^\circ \in {\mathcal {D}}({\mathbb {T}})$ is said to converge to $\Phi ^\circ \in {\mathcal {D}}({\mathbb {T}})$ if, for every natural number k, the functions $D^k\Phi _m^\circ$ converge uniformly to $D^k\Phi ^\circ$.

Fig. 3.3

Periodic function versus function on ${\mathbb {T}}$

In Sect. 3.2 we saw that every distribution T can be generated as the limit of a sequence of regular distributions $f_m$. Applying this result

$$\begin{aligned} \langle T,\phi \rangle = \lim _{m\rightarrow \infty } \langle f_m,\phi \rangle = \lim _{m\rightarrow \infty } \int \limits _{\mathbb {T}}f_m^\circ (p) \, \Phi ^\circ (p) \, dp = \langle T^\circ ,\Phi ^\circ \rangle \end{aligned}$$

we see that not only regular, but every periodic distribution can be equivalently represented by a continuous, linear functional on ${\mathcal {D}}({\mathbb {T}})$.

To see the converse, that is that every continuous, linear functional on ${\mathcal {D}}({\mathbb {T}})$ represents a distribution on ${\mathcal {D}}({\mathbb {R}})$, we have to show that every indefinitely differentiable ${\mathcal {T}}$-periodic function $\Phi$ can be generated by some test function $\phi$ as in (3.24). To this end we introduce the so-called unitary functions. These are test functions for which there is a number ${\mathcal {T}}$ such that

$$\begin{aligned} \sum _{m=-\infty }^\infty \xi (t - m{\mathcal {T}}) = 1 . \end{aligned}$$

(3.26)

Note that, here again, the sum is finite for every bounded range of t. We can find several such functions. The following example satisfies (3.26) for ${\mathcal {T}}= 1$ (see Fig. 3.4)

With it we can construct unitary functions for arbitrary periods ${\mathcal {T}}$ by $\xi _{\mathcal {T}}(t) = \xi _1(t/{\mathcal {T}})$.

Fig. 3.4

Example unitary test function

Now, given any ${\mathcal {T}}$-periodic test function $\Phi$ and an unitary function $\xi _{\mathcal {T}}$ we have

$$\begin{aligned} \Phi (t) = & {} \Phi (t) \sum _{m=-\infty }^\infty \xi _{\mathcal {T}}(t - m{\mathcal {T}}) \nonumber \\ = & {} \sum _{m=-\infty }^\infty \xi _{\mathcal {T}}(t - m{\mathcal {T}}) \, \Phi (t - m{\mathcal {T}}) . \end{aligned}$$

(3.27)

Since $\xi _{\mathcal {T}}$ is a test function, so is $\xi _{\mathcal {T}}\, \Phi$. We have thus established that every indefinitely differentiable periodic function $\Phi$ can be represented as the sum of a test function $\phi$. We conclude that periodic distributions are in one-to-one correspondence with continuous, linear functionals on ${\mathcal {D}}({\mathbb {T}})$. It is therefore natural to call these functionals distributions on ${\mathbb {T}}$. They form a vector space that is denoted by  ${\mathcal {D'}}({\mathbb {T}})$.

We now have a second way to define periodic distributions.

Definition 3.5

(Periodic distribution II) A periodic distribution T is defined by

$$\begin{aligned} \langle T,\phi \rangle :=\langle T^\circ ,\Phi ^\circ \rangle \end{aligned}$$

(3.28)

with $T^\circ$ a distribution in ${\mathcal {D'}}({\mathbb {T}})$ and where $\phi$ and $\Phi ^\circ$ are related by Eqs. (3.24) and (3.25).

This definition is compatible with the first one since replacing $\phi (t)$ by $\phi (t + {\mathcal {T}})$ doesn’t change $\Phi ^\circ$.

Example 3.7: Dirac comb $\boldsymbol{\delta _{\mathcal {T}}}$

Consider the distribution in ${\mathcal {D'}}({\mathbb {T}})$ defined by a Dirac pulse $\delta ^\circ$ with support consisting of the point at arc-length $p=0$. Its value on a test function in ${\mathcal {D}}({\mathbb {T}})$ is

$$\begin{aligned} \langle \delta ^\circ (p),\Phi ^\circ (p) \rangle = \Phi ^\circ (0)\,. \end{aligned}$$

The corresponding periodic distribution in ${\mathcal {D'}}({\mathbb {R}})$ is 

$$\begin{aligned} \delta _{\mathcal {T}}(t) :=\sum _{m=-\infty }^\infty \delta (t + m{\mathcal {T}}) \end{aligned}$$

and evaluates to the same value as $\delta ^\circ$

$$\begin{aligned} \langle *,\sum _{m=-\infty }^\infty \delta (t + m{\mathcal {T}}) \rangle {\phi (t)} = & {} \sum _{m=-\infty }^\infty \langle \delta (t + m{\mathcal {T}}),\xi _{\mathcal {T}}(t) \, \Phi (t) \rangle \\ = & {} \sum _{m=-\infty }^\infty \xi _{\mathcal {T}}(-m{\mathcal {T}}) \, \Phi (-m{\mathcal {T}}) \\ = & {} \Phi ^\circ (0) \sum _{m=-\infty }^\infty \xi _{\mathcal {T}}(-m{\mathcal {T}})\\ = & {} \Phi ^\circ (0)\,. \end{aligned}$$

Since the support of distributions in ${\mathcal {D'}}({\mathbb {T}})$ is bounded, with the second definition, the convolution of periodic distributions is always well-defined and associative

$$\begin{aligned} \langle S *T,\phi \rangle = & {} \langle S(\tau ) \otimes T(\lambda ),\phi (\tau + \lambda ) \rangle \\ = & {} \langle S^\circ (\tau ),\langle T^\circ (\lambda ),\Phi ^\circ (\tau + \lambda ) \rangle \rangle \\ = & {} \langle S^\circ *T^\circ ,\Phi ^\circ \rangle . \end{aligned}$$

In addition it’s easily verified by replacing $\phi (t)$ by $\phi (t + {\mathcal {T}})$ that the resulting distribution is also ${\mathcal {T}}$-periodic. In other words, ${\mathcal {D'}}({\mathbb {T}})$ is closed under convolution.