Distributions

Abstract

We investigate the mathematical description of signals that are commonly used in the analysis of technical problems. From a mathematical point of view it would be useful to limit the signals of interest to the set of continuous functions. However, this has several disadvantages. For example, suppose we are interested in the transient response of, say, a series RC low-pass-filter (LPF) to an input step voltage.

We investigate the mathematical description of signals that are commonly used in the analysis of technical problems. From a mathematical point of view it would be useful to limit the signals of interest to the set of continuous functions. However, this has several disadvantages. For example, suppose we are interested in the transient response of, say, a series RC low-pass-filter (LPF) to an input step voltage. If we describe the input signal with a continuous function, then the details of the calculations depend on the chosen description of the input signal rise transient. This however tends to mask the fact that, if the LPF time constant is much larger than the input signal rise-time, the output response is essentially independent of the input signal transient shape. For this reason, in these situations, it is much more convenient to use an idealized input unit step function such as the Heaviside unit step function

$$\begin{aligned} \textsf{1}_{+}(t) = \left\{ \begin{array}{ll} 0 &{} t < 0 \\ 1 &{} t \ge 0 \end{array} \right. \end{aligned}$$

which is not continuous at $t=0$.

Consider further the LPF example. If we write the differential equation using the current as unknown, then we need the derivative of the driving signal. However, the derivative of the function $\textsf{1}_{+}$ does not exist at $t=0$ and is zero at every other point. We are therefore led to introduce the so-called Dirac impulse ${\delta }$ which however is not even a function, but rather a generalized function or distribution.

It follows from these considerations that a correct description of commonly used signals belongs to the theory of distributions. Distributions have many useful properties. The key one being that they can be differentiated any number of times. The main contributor to the development of this theory was Schwartz [16].

1 Test Functions

The key idea in the theory of distributions, is not to direct attention to the value of a function at every point of its domain, but instead to “measure” the behavior of a function when acting on a class of particularly well-behaved functions. In this section we introduce one such class of functions, the class of test functions.

Let $k=(k_1,\ldots ,k_n)$ be an n-tuple of non-negative integers called a multi-index. The  differential operator of order |k| is defined by

$$\begin{aligned} D^k:=D_1^{k_1} \cdot \cdots \cdot D_n^{k_n}\,, \quad D_i:=\frac{\partial }{\partial \tau _i} \end{aligned}$$

(2.1)

with

$$\begin{aligned} |k| :=k_1 + \cdots + k_n \end{aligned}$$

(2.2)

the length of the multi-index k and $\tau \in {\mathbb {R}}^n$. For functions of a single variable we also use the following shorter notation for the kth order derivative

$$\begin{aligned} f^{(k)} :=\frac{{\textrm{d}}^k f}{{\textrm{d}}\tau ^k} \end{aligned}$$

(2.3)

where in this case k is of course a single non-negative integer.

Given an open set $U \subset {\mathbb {R}}^n$, the set of all k-times continuously differentiable functions $f: U \rightarrow {\mathbb {C}}$ is denoted by ${\mathcal {C}}^k(U)$ or simply ${\mathcal {C}}^k$.

Definition 2.1

(Test function) A function $\phi : {\mathbb {R}}^n \rightarrow {\mathbb {C}}$ is called a test function if it is indefinitely differentiable and has compact support, that is, if $\phi \in {\mathcal {C}}^\infty$ and $\phi (\tau _1,\ldots ,\tau _n) = 0$ outside a compact set K. The vector space of all such functions is denoted by ${\mathcal {D}}$.

To define a continuity criterion for distributions we need to define a topology that can be encoded in the form of a convergence principle in ${\mathcal {D}}$.

Definition 2.2

(Convergence of test functions) A  sequence of functions $\phi _m \in {\mathcal {D}}, m\in {\mathbb {N}}$ is said to converge to $\phi \in {\mathcal {D}}$, in symbols

$$\begin{aligned} \phi _m \ \underset{{\mathcal {D}}}{\longrightarrow }\ \phi \quad \text {or}\quad \lim _{m\rightarrow \infty }\phi _m=\phi \,, \end{aligned}$$

if the following two conditions are met:

• 1. There exist a compact set K such that it includes the support of all $\phi _m$ and of $\phi$.

• 2. For every n-tuple k the sequence of functions $D^k\phi _m$ converges uniformly toward $D^k\phi$.

Fig. 2.1

Example test function

These conditions ensure that the limiting function (i) has compact support and (ii) that it is indefinitely differentiable, in other words, that the limiting function is also a test function.

Example 2.1: Test function

Consider the following functions (see Fig. 2.1)

$$\begin{aligned} \beta _\nu (t) := & {} \left\{ \begin{array}{ll} \frac{\nu }{B}{\textrm{e}}^{\frac{-1}{1-(\nu t)^2}} &{} \textrm{for}\ |\nu t|\, < 1 \\ 0&{} \textrm{for}\ |\nu t|\, \ge 1 \end{array}\right. \\ B := & {} \int _{-1}^1 {\textrm{e}}^{\frac{-1}{1-t^2}} \,{\textrm{d}}t \nonumber \end{aligned}$$

(2.4)

For each value of $\nu > 0$ and for $|\nu t| < 1$ the function $\beta _\nu$ is the composition of a rational function with no singularities and the exponential function. Since the latter two functions are indefinitely differentiable and the composition of indefinitely differentiable functions is also indefinitely differentiable, it follows that, in this range, $\beta _\nu$ is indefinitely differentiable.

To establish that $\beta _\nu$ is a test function we have further to show that

$$\begin{aligned} \lim _{|\nu t|\uparrow 1} D^k\beta _\nu (t) = 0 \end{aligned}$$

(2.5)

for all values of k. This can be done by induction: assume that the kth order derivative is the product of $\beta _\nu$ and a polynomial in the two variables $\tau _1 = 1/(1-\nu t)$ and $\tau _2 = 1/(1+\nu t)$

$$\begin{aligned} D^k\beta _\nu (t) = p_k\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) \beta _\nu (t) \ . \end{aligned}$$

(2.6)

This is clearly the case for $k=0$. We show that this is then true for $k+1$:

$$\begin{aligned} &{D^{k+1}\beta _\nu (t) } \nonumber \\ = & {} \left[ - \frac{\nu \,(D_2p_k)\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) }{\left( \nu \,t+1\right) ^2} + \frac{\nu \,p_k\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) }{2\,\left( \nu \,t+1\right) ^2} \right. \nonumber \\ {} & {} \left. + \frac{\nu \,(D_1p_k)\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) }{\left( \nu \,t-1\right) ^2} - \frac{\nu \,p_k\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) }{2\,\left( \nu \,t-1\right) ^2} \right] \,\beta _\nu (t) \nonumber \\ =: & {} p_{k+1}\left( \frac{1}{1-\nu t},\frac{1}{1+\nu t}\right) \beta _\nu (t) \ . \end{aligned}$$

(2.7)

If we express the limit as $\nu t$ tends to 1 in terms of $\tau _1$, we see that it is the limit of the product of a polynomial and a decreasing exponential which converges to 0

$$\begin{aligned} \lim _{\nu t\uparrow 1} D^k\beta _\nu (t) = \lim _{\tau _1 \rightarrow \infty }p_k(\tau _1,\frac{\tau _1}{2\tau _1-1})\, \frac{\nu }{B}\,e^{-\frac{\tau _1^2}{2\tau _1-1}} = 0 \ . \end{aligned}$$

(2.8)

Similarly, the limit towards –1 can be expressed in terms of $\tau _2$ with the same result. Hence $\beta _\nu \in {\mathcal {C}}^\infty$.

While $\beta _\nu$ is a test function for each value of $\nu$, the sequence $(\beta _m), m \in {\mathbb {N}}$ doesn’t converge in ${\mathcal {D}}$. For $t \ne 0, m \rightarrow \infty$ the value of $\beta _m(t)$ converges toward zero, while the value of the functions at $t = 0$ grows without bounds. The limiting function is therefore not continuous.

The sequence $(\beta _{1/m})$ also doesn’t converge in ${\mathcal {D}}$. As $m \rightarrow \infty$ the support of the functions grows without bounds. It is therefore not possible to find a compact set K containing the support of all members of the sequence as well as that of the limiting function.

An example of a converging sequence is $\beta _m/m^2$ which converges toward the zero function.

Fig. 2.2

a Regularized of the discontinuous function $\textsf{1}_{2}(.)$ b Construction of the regularized of $\textsf{1}_{2}(.)$

Example 2.2: Regularisation

Consider an impulse of finite duration (see Fig. 2.2a)

$$\begin{aligned} \textsf{1}_{k}(t) :=\textsf{1}_{+}(t) - \textsf{1}_{+}(t-k) = \left\{ \begin{array}{ll} 1 &{} 0 \le t < k\\ 0 &{} \textrm{otherwise} \end{array} \right. \end{aligned}$$

This function is clearly not continuous at $t=0$ and $t=k$. These jump discontinuities can be removed by convolving $\textsf{1}_{k}$ with the function $\beta _\nu$ of the previous example

$$\begin{aligned} \textsf{1}_{k}*\beta _\nu (t) = \int _{-\infty }^{\infty }\textsf{1}_{k}(\tau )\,\beta _\nu (t-\tau )\,d\tau = \int _0^k\beta _\nu (t-\tau )\,d\tau \ . \end{aligned}$$

(2.9)

We say that the so obtained function is the regularised of $\textsf{1}_{k}$ by $\beta _\nu$ (see Fig. 2.2a).

Observe that $\textsf{1}_{k}*\beta _\nu$ is just a definite integral of $\beta _\nu$ and is therefore indefinitely differentiable. If the support of $\beta _\nu$ lies completely within the integration range, then, given the chosen normalization constant for $\beta _\nu$, the value of $\textsf{1}_{k}*\beta _\nu$ is 1. If the support of $\beta _\nu$ doesn’t intersect the integration range, then the value of $\textsf{1}_{k}*\beta _\nu$ is 0. For the remaining values of the independent variable t, $0 < \textsf{1}_{k}*\beta _\nu (t) < 1$ (see Fig. 2.2b)

$$\begin{aligned} \textsf{1}_{k}*\beta _\nu (t) = \left\{ \begin{array}{ll} 1 &{} 1/\nu \le t \le k-1/\nu \\ 0 &{} t \le -1/\nu \ \textrm{or}\ t \ge k+1/\nu \\ > 0\ \textrm{and} < 1 &{} \mathrm {otherwise.} \end{array} \right. \end{aligned}$$

(2.10)

We have thus established that $\textsf{1}_{k}*\beta _\nu \in {\mathcal {D}}$.

From this example we see that for any open interval U and any closed interval $K \subset U$ we can construct a real valued test function $\phi$ with $0 \le \phi (t) \le 1$, a value of 1 within K and a value of 0 outside of U. This is a useful property that we will exploit later.

A similar construction can be made for test functions of more than one variable. For later reference we define a real valued test function with values between 0 and 1 that we call  $\alpha$ such that

$$\begin{aligned} \alpha : {\mathbb {R}}^n \rightarrow [0,1],\quad \tau \mapsto \left\{ \begin{array}{ll} 1 &{} |\tau | \le 1\\ 0 &{} |\tau | \ge 2 \,. \end{array} \right. \end{aligned}$$

(2.11)

2 Distributions

A key aspect of the theory of distributions is the fact that it makes continuous functions differentiable any number of times. To see how this goes, remember from calculus that by partial integration we can transfer the operation of differentiation from one function to another one. Thus, if we pair the function of interest f with a function $\phi$ differentiable everywhere, then we can relate the derivative of f with a well-defined expression

$$\begin{aligned} \int _{-\infty }^\infty Df(\tau )\, \phi (\tau )\, d\tau = f(\tau )\,\phi (\tau )\arrowvert _{-\infty }^\infty \, - \int _{-\infty }^\infty f(\tau )\, D\phi (\tau )\, d\tau \ . \end{aligned}$$

(2.12)

To make the expression independent of the limits of integration, the first term on the right-hand side should disappear. This can be achieved, for example, by choosing a function $\phi$ with compact support. In addition, to be able to assign a meaning to the derivative of any order, the function $\phi$ should be indefinitely differentiable. Note that these are precisely the properties of test functions.

An additional requirement is that of the assignment being unique. For example, the right-hand side expression should be identically zero only if $Df = 0$ (almost everywhere). Suppose that f has compact support. If the support of $D\phi$ doesn’t overlap with the one of f then the right-hand expression is also zero and the assignment is not unique. To avoid this situation we are forced to pair the function f with every test function $\phi \in {\mathcal {D}}$.

A distribution is a generalization of these ideas and is defined as follows.

Definition 2.3

(Distribution) A distribution is defined as a linear, continuous function on the set of test functions

$$\begin{aligned} T: {\mathcal {D}}({\mathbb {R}}^n) \rightarrow {\mathbb {C}}\ , \quad \phi \mapsto \langle T,\phi \rangle \end{aligned}$$

(2.13)

This means that a distribution T has the following properties:

• 1. $\langle T,\phi _1 + \phi _2 \rangle = \langle T,\phi _1 \rangle + \langle T,\phi _2 \rangle$ for all $\phi _1,\, \phi _2 \in {\mathcal {D}}$.

• 2. $\langle T,c\,\phi \rangle = c\,\langle T,\phi \rangle$ for all $\phi \in {\mathcal {D}}$ and $c\in {\mathbb {C}}$.

• 3. From $\phi _k\ \underset{{\mathcal {D}}}{\longrightarrow }\ \phi$ it follows that $\langle T,\phi _k \rangle \rightarrow \langle T,\phi \rangle$, where the latter is the normal convergence of complex numbers.

Since distributions are linear by definition, the condition of continuity can be expressed in a slightly different, but equivalent way:

• 3’. From $\phi _k\ \underset{{\mathcal {D}}}{\longrightarrow }\ 0$ it follows that $\langle T,\phi _k \rangle \rightarrow 0$.

Two distributions $T_1$ and $T_2$ are equal if $\langle T_1,\phi \rangle = \langle T_2,\phi \rangle$ for every test function $\phi \in {\mathcal {D}}$. A distribution is called realif it evaluates to a real number when applied to any real valued test function.

The set of all distributions forms a vector space denoted by  ${\mathcal {D'}}$, where addition of two distributions $T_1$ and $T_2$ and multiplication with a complex constant c are defined by

$$\begin{aligned} \langle T_1+T_2,\phi \rangle := & {} \langle T_1,\phi \rangle + \langle T_2,\phi \rangle \\ \langle c T,\phi \rangle := & {} c \langle T,\phi \rangle = \langle T,c\,\phi \rangle . \end{aligned}$$

A mapping assigning a number to every element of a vector space is called a functional. Distributions are therefore functionals on test functions.

Example 2.3: Functions as distributions

Consider a continuous function $f \in {\mathcal {C}}({\mathbb {R}}^n)$. We can associate with it a distribution $T_f$ by the procedure outlined at the beginning of the section

$$\begin{aligned} \langle T_f,\phi \rangle = \int _{{\mathbb {R}}^n} f(\tau )\,\phi (\tau )\,d^n\tau \,. \end{aligned}$$

(2.14)

Linearity is clear from the properties of integrals. To see that it is continuous, consider a sequence of test functions converging to zero $\phi _m\ \underset{{\mathcal {D}}}{\longrightarrow }\ 0$. Then

$$\begin{aligned} \int _{{\mathbb {R}}^n} f(\tau )\,\phi _m(\tau )\,d^n\tau \le \sup _{t\in K}|\phi _m(t)|\int _K |f(\tau )|\,d^n\tau \ \underset{}{\longrightarrow }\ 0 \end{aligned}$$

with K a compact set including the support of all $\phi _m$.

Consider now two continuous functions $f_1$ and $f_2$. If $\langle T_{f_1},\phi \rangle = \langle T_{f_2},\phi \rangle$ for every $\phi \in {\mathcal {D}}$, then, by the properties of integrals of continuous functions, it follows that $f_1 = f_2$. We thus have an injective mapping from continuous functions to distributions. We can therefore identify continuous functions with their corresponding distributions and write $\langle f,\phi \rangle$ instead of $\langle T_f,\phi \rangle$.

The theory of distributions requires the use of Lebesgue integrals as opposed to Riemann ones, as Lebesgue’s integration theory is more powerful and allows integrating a broader set of functions. Of course, when both integrals do exist, they coincide. A key concept in the Lebesgue theory of integration is that of the measure. For our purposes we can think of the  Lebesgue measure as a volume and a set of zero measure in ${\mathbb {R}}^n$ as a (sufficiently regular) subspace of dimension $k<n$. A point on the real line ${\mathbb {R}}$, a line on a plane and a surface in ${\mathbb {R}}^3$ are all examples of sets of zero measure. The union of a denumerable family of sets of zero measure is itself a set of zero measure. Therefore, the set of rational numbers on the real line ${\mathbb {R}}$ has zero measure. Two locally integrable functions differing only on a set of zero measure are said to be equal almost everywhere.

Example 2.4: Locally integrable functions

Consider a locally integrablefunction $f \in {\mathcal {L}}_{\textrm{loc}}^1({\mathbb {R}}^n)$, a function that is Lebesgue integrable over every compact set $K\subset {\mathbb {R}}^n$. As in the previous example we can associate it with a distribution through the integral (2.14). In this case however the mapping is not injective. Any two locally integrable functions $f_1$ and $f_2$ differing only in a set of measure zero produce the same value $\langle f_1,\phi \rangle = \langle f_2,\phi \rangle$ for every $\phi \in {\mathcal {D}}$. That means that they map to the same distribution.

In physical and engineering applications the values of a function in a set of zero measure is often unimportant. It is therefore natural to consider the equivalence class of all functions differing at most on a set of zero measure. In this way we obtain again an injective mapping, but now from the equivalence class of locally integrable functions differing at most on a set of zero measure (equal almost everywhere) to distributions, and we can again identify without ambiguity the former with the latter. To avoid overloading the notation we write a representative for the equivalence class, that is, we write $\langle f,\phi \rangle$ where f is a representative.

All distributions that can be represented by locally integrable functions through (2.14) are called regular distributions. However, not all distributions are regular and distributions that aren’t regular are called singular distributions. Nonetheless, regular distributions are dense in ${\mathcal {D'}}$. That is, in a similar way as real numbers arise as a limiting process from rational ones, any distribution can be represented as a limit of regular distributions, where the convergence of distributions is defined as follows.

Definition 2.4

(Convergence of distributions) A sequence of distributions $(T_m)_{m\in {\mathbb {N}}}$ is said to converge to the distribution T, if the sequence of numbers $\langle T_m,\phi \rangle$ converges to the number $\langle T,\phi \rangle$ for every $\phi \in {\mathcal {D}}$. In symbols

$$\begin{aligned} T_m \ \underset{{\mathcal {D'}}}{\longrightarrow }\ T\quad \text {or}\quad \lim _{m\rightarrow \infty }T_m=T \end{aligned}$$

if

$$\begin{aligned} \langle T_m,\phi \rangle \ \underset{}{\longrightarrow }\ \langle T,\phi \rangle \quad \text {for every } \phi \in {\mathcal {D}}\,. \end{aligned}$$

It is not obvious that the limit T is in fact a distribution, that is, linear and continuous. However, this is indeed the case. The space ${\mathcal {D'}}$ is thus closed under convergence. A proof can be found in [18].

This definition is based on a discrete parameter m traversing the natural numbers. If the parameter traverses a continuous set of values the situation is similar and can be reduced to the discrete case. Consider the sequence of distributions $T_\nu$ depending on the continuous parameter $\nu \in {\mathbb {R}}$. For each value of $\nu$ and each test function $\phi$, the functional $\langle T_\nu ,\phi \rangle$ evaluates to a number. For each test function $\phi$ the set of distributions $T_\nu$ therefore defines a function of $\nu$

$$\begin{aligned} \zeta (\nu ) = \langle T_\nu ,\phi \rangle . \end{aligned}$$

Lets define a sequence $(\nu _m)_{m \in {\mathbb {N}}}$ of values converging toward infinity. If for every such sequence and every test function

$$\begin{aligned} \lim _{k \rightarrow \infty } \zeta (\nu _k) = \lim _{k \rightarrow \infty } \langle T_{\nu _k},\phi \rangle = \langle T,\phi \rangle \end{aligned}$$

then

$$\begin{aligned} \lim _{\nu \rightarrow \infty } \langle T_{\nu },\phi \rangle = \langle T,\phi \rangle . \end{aligned}$$

Similarly for a continuous parameter converging toward a finite limit $\eta$.

Example 2.5: Dirac delta distribution

Consider the functions $\beta _m$ of Example 2.1. The regular distributions associated with these functions form a sequence converging to a singular distribution. We have

$$\begin{aligned} &{\lim _{m\rightarrow \infty }\langle \beta _m,\phi \rangle }\\ &= \lim _{m\rightarrow \infty }\int _{-1/m}^{1/m}\beta _m(\tau )\,\phi (\tau )\,d\tau \\ &= \lim _{m\rightarrow \infty }\left\{ \int _{-1/m}^{1/m}\beta _m(\tau )\,\phi (0)\,d\tau + \int _{-1/m}^{1/m} \beta _m(\tau )\,[\phi (\tau ) - \phi (0)]\,d\tau \right\} \end{aligned}$$

Since test functions are continuous and differentiable we can use the mean value theorem to express $\phi$ as

$$\begin{aligned} \phi (\tau ) = \phi (0) + D\phi (\lambda )\,\tau \end{aligned}$$

for some $\lambda \in (0,\tau )$. With this we can see that the second term converges to zero

$$\begin{aligned} \int _{-1/m}^{1/m}\beta _m(\tau )\,[\phi (\tau ) - \phi (0)]\,d\tau {} & {} \le \int _{-1/m}^{1/m}|\beta _m(\tau )|\,|\phi (\tau ) - \phi (0)|\,d\tau \\ {} & {} \le \sup _{\lambda \in (-\frac{1}{m},\frac{1}{m})}\frac{|D\phi (\lambda )|}{m} \int _{-1/m}^{1/m}\,|\beta _m(\tau )|\,d\tau \\ {} & {} \ \underset{}{\longrightarrow }\ 0 \end{aligned}$$

We therefore obtain

$$\begin{aligned} \lim _{m\rightarrow \infty }\langle \beta _m,\phi \rangle = & {} \phi (0)\,\lim _{m\rightarrow \infty }\int _{-1/m}^{1/m}\beta _m(\tau )\,d\tau \\ = & {} \phi (0)\,\lim _{m\rightarrow \infty } 1 \\ = & {} \phi (0) . \end{aligned}$$

The sequence $\beta _m$ thus converge to the Dirac delta distribution ${\delta }$ which is defined by

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

(2.15)

Besides the sequence $\beta _m$ there are many other regular distribution sequences converging to ${\delta }$. For example, with the same procedure used above, it is simple to show that the sequence defined by the following functions does also converge to ${\delta }$

$$\begin{aligned} f_m(t) = \left\{ \begin{array}{ll} m/2 &{} |t| \le 1/m \\ 0 &{} |t| > 1/m \end{array} \right. \end{aligned}$$

Note that the notation used in many technical texts to define the Dirac delta distribution is not mathematically correct and only has a symbolic value

$$\begin{aligned} \int _{-\infty }^\infty {\delta }(\tau )\,\phi (\tau )\,d\tau = \phi (0). \end{aligned}$$

This notation imply the existence of a function with a value of zero everywhere but at $\tau = 0$ where its value is infinite. However, the value of the Lebesgue integral of such a function is zero since a single point of the real line has zero measure. This notation is however useful as it helps to remember several properties that we will see shortly.

Example 2.6: Cauchy principal value

The function $f(\tau )=1/\tau$ is not locally integrable. For this reason we can’t associate with it a regular distribution through (2.14). A way around this is to use the  Cauchy principal value of the integral to define the following singular distribution

$$\begin{aligned} \Big \langle \textrm{pv}\,\frac{1}{\tau },\phi \Big \rangle := & {} \textrm{pv}\int _{-\infty }^\infty \frac{\phi (\tau )}{\tau }\,d\tau \nonumber \\ = & {} \lim _{\epsilon \downarrow 0}\left\{ \int _{-\infty }^{-\epsilon } \frac{\phi (\tau )}{\tau }\,d\tau + \int _\epsilon ^\infty \frac{\phi (\tau )}{\tau }\,d\tau \right\} \end{aligned}$$

(2.16)

Integrating by parts the first integral we obtain

$$\begin{aligned} \int _{-\infty }^{-\epsilon } \frac{\phi (\tau )}{\tau }\,d\tau = & {} \left. \left( \phi (\tau )\,\ln |\tau |\right) \right| _{-\infty }^{-\epsilon } - \int _{-\infty }^{-\epsilon } \ln |\tau |\,D\phi (\tau )\,d\tau \\ = & {} \phi (-\epsilon )\,\ln |\epsilon | - \int _{-\infty }^{-\epsilon } \ln |\tau |\,D\phi (\tau )\,d\tau \end{aligned}$$

and similarly for the second integral

$$\begin{aligned} \int _{\epsilon }^{\infty }\frac{\phi (\tau )}{\tau }\,d\tau = & {} -\phi (\epsilon )\,\ln (\epsilon ) - \int _{\epsilon }^{\infty }\ln (\tau )\,D\phi (\tau )\,d\tau \,. \end{aligned}$$

We see that the first term of both integrals do diverge as $\epsilon$ goes to 0. However, using the mean value theorem, we note that there are values $\lambda _1\in (0,\epsilon )$ and $\lambda _2\in (-\epsilon ,0)$ such that

$$\begin{aligned} \phi (\epsilon ) = & {} \phi (0) + \epsilon D\phi (\lambda _1)\\ \phi (-\epsilon ) = & {} \phi (0) - \epsilon D\phi (\lambda _2) \end{aligned}$$

With $M=-\left( D\phi (\lambda _1) + D\phi (\lambda _2)\right)$, the limit of the sum of the diverging parts therefore do cancel

$$\begin{aligned} \lim _{\epsilon \downarrow 0}\,M\,\epsilon \,\ln |\epsilon | = & {} 0 \end{aligned}$$

and we finally obtain

$$\begin{aligned} \langle \textrm{pv}\,\frac{1}{\tau },\phi \rangle = & {} -\int _{-\infty }^{\infty }\ln |\tau |\,D\phi (\tau )\,d\tau \,. \end{aligned}$$

This last integral is well defined as $\ln |\tau |$ is locally integrable and therefore defines a well defined regular distribution. We will meet this distribution again in the context of the Fourier transform of distributions.

3 Basic Properties

There are some useful operations that we can perform on locally integrable functions that can be carried over to distributions. A common operation is to shift a function f by an amount $\tau$ to obtain $t \mapsto f(t-\tau )$. If we apply the change of variable $\lambda =t-\tau$ to the regular distribution associated with the shifted function we obtain

$$\begin{aligned} \int _{-\infty }^\infty f(t-\tau )\,\phi (t)\,d t = & {} \int _{-\infty }^\infty f(\lambda )\,\phi (\lambda +\tau )\,d\lambda . \end{aligned}$$

By generalizing this result we define the operation of shifting a distribution by

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

(2.17)

With this definition we can for example denote a Dirac pulse at time $\tau$ by ${\delta }(t-\tau )$

$$\begin{aligned} \langle {\delta }(t-\tau ),\phi (t) \rangle := & {} \phi (\tau ) . \end{aligned}$$

Notation

Note that a distribution T isn’t a function of the variable t. In spite of this it is useful to write T(t) to indicate the symbol used for the independent variable of the testing function (this will be useful when we’ll introduce operations such as the convolution) and as a convenient notation to indicate some operations such as shifting. In no way this is meant to imply the existence of a function or that the distribution is regular.

Another useful operation is multiplication of the independent variable of a function by a constant a. By generalizing what happens with regular distributions, we define multiplication of the independent variable by a constant a for any distribution in ${\mathcal {D'}}({\mathbb {R}}^n)$ by

$$\begin{aligned} \langle T(a\,t),\phi (t) \rangle := & {} \Big \langle T(t),\frac{1}{|a|^n}\phi \left( \frac{t}{a}\right) \Big \rangle \,. \end{aligned}$$

(2.18)

This operation is closely related to the concepts of even and odd distributions.

Definition 2.5

(Even and odd distributions) An even distribution T is defined as a distribution for which, for every test function $\phi$

$$\begin{aligned} \langle T(t),\phi (-t) \rangle = \langle T(t),\phi (t) \rangle . \end{aligned}$$

(2.19)

Similarly, an odd distribution satisfies

$$\begin{aligned} \langle T(t),\phi (-t) \rangle = - \langle T(t),\phi (t) \rangle \end{aligned}$$

(2.20)

for every test function $\phi$.

A further useful operation is multiplication of a distribution with an indefinitely differentiable function $\gamma$. First note that multiplication of a test function $\phi$ with an indefinitely differentiable function results in another test function. For this reason we can again generalize the behavior of regular distributions and define

$$\begin{aligned} \langle \gamma \,T,\phi \rangle := & {} \langle T,\gamma \,\phi \rangle \,. \end{aligned}$$

(2.21)

4 Differentiation of Distributions

At the beginning of Sect. 2.2 we mentioned that one of the distinguishing features of distributions is the fact that they can be differentiated any number of times. We also argued that, for regular distributions, partial integration leads to an expression which can be considered as the definition of the derivative of regular distributions

$$\begin{aligned} \langle f^{(1)},\phi \rangle = & {} \int _{-\infty }^\infty f^{(1)}(\tau )\, \phi (\tau )\, d\tau \nonumber \\ = & {} - \int _{-\infty }^\infty f(\tau )\, \phi ^{(1)}(\tau )\, d\tau \nonumber \\ = & {} \langle f,-\phi ^{(1)} \rangle . \end{aligned}$$

(2.22)

In fact, this definition can be extended to singular distributions and to distributions of several variables, that is to arbitrary distributions.

Definition 2.6

The first order partial derivative of a distribution T on ${\mathcal {D}}({\mathbb {R}}^n)$ is defined by

$$\begin{aligned} \langle D_i T,\phi \rangle :=\langle T,- D_i\phi \rangle \quad i = 1,\ldots , n\,. \end{aligned}$$

(2.23)

Since the derivative of a test function $D_i\phi$ is still a test function, it follows that the derivative of a distribution is always a distribution and that distributions can be differentiated an arbitrary number of times.

With k an n-tuple of non-negative integers, the derivative of order |k| follows from the above definition

$$\begin{aligned} \langle D^k T,\phi \rangle = (-1)^{|k|}\langle T,D^k\phi \rangle \,. \end{aligned}$$

(2.24)

The order of differentiation is irrelevant since test functions have continuous partial derivatives of all orders and hence

$$\begin{aligned} \langle D_iD_jT,\phi \rangle = \langle T,D_jD_i\phi \rangle = \langle T,D_iD_j\phi \rangle = \langle D_jD_iT,\phi \rangle \,. \end{aligned}$$

The rule for the differentiation of the product of a distribution T and an indefinitely differentiable function $\gamma$ is the same as the rule of differentiation for the product of two functions

$$\begin{aligned} \langle D_i(\gamma T),\phi \rangle = & {} -\langle \gamma T,D_i\phi \rangle = -\langle T,\gamma \,D_i\phi \rangle \\ = & {} -\langle T,D_i(\gamma \,\phi ) \rangle + \langle T,D_i\gamma \,\phi \rangle \\ = & {} \langle D_i T,\gamma \,\phi \rangle + \langle D_i\gamma \,T,\phi \rangle \\ = & {} \langle \gamma \,D_i T,\phi \rangle + \langle D_i\gamma \,T,\phi \rangle \end{aligned}$$

or

$$\begin{aligned} D_i(\gamma T) = \gamma \,D_i T + (D_i\gamma )\,T \,. \end{aligned}$$

(2.25)

Two important properties of distributional differentiation follow immediately from the definition. The first is that differentiation is a linear operation: given two distributions $T_1$ and $T_2$ and two numbers $c_1$ and $c_2$

$$\begin{aligned} D^k(c_1\,T_1 + c_2\,T_2) = c_1\,D^kT_1 + c_2\,D^kT_2 \,. \end{aligned}$$

(2.26)

The second is continuity: given a sequence of distributions $(T_m)_{m\in {\mathbb {N}}}$ converging toward a distribution T, the sequence of corresponding partial derivatives $(D^kT_m)_{m\in {\mathbb {N}}}$ converges to $D^kT$

$$\begin{aligned} \lim _{m\rightarrow \infty } \langle D^k T_m,\phi \rangle = & {} \lim _{m\rightarrow \infty } (-1)^{|k|}\langle T_m,D^k\phi \rangle \nonumber \\ = & {} (-1)^{|k|}\langle T,D^k\phi \rangle \nonumber \\ = & {} \langle D^k T,\phi \rangle \,. \end{aligned}$$

(2.27)

In other words, the operations of limit-taking and differentiation can always be exchanged. In particular this means that if a sequence of partial sums $S_m= \sum _{i=0}^{m-1}\,T_i$ converges to a series $S = \sum _{i=0}^\infty \,T_i$, then the series can be differentiated term by term.

Notation

To distinguish a regular distribution defined by the usual derivative of a function f from the derivative in the sense of distributions of the regular distribution defined by f, we are going to always denote the former by $T_{f^{(k)}}$ or $T_{D^k f}$. The later will be denoted interchangeably by $f^{(k)}$, $D^k f$, $T_f^{(k)}$ or $D^kT_f$.

Example 2.7: Derivative of $\boldsymbol{{\delta }}$

The first order derivative of the Dirac delta distribution is

$$\begin{aligned} \langle {\delta }^{(1)},\phi \rangle = -\langle {\delta },\phi ^{(1)} \rangle = -\phi ^{(1)}(0) \,. \end{aligned}$$

The kth order one is

$$\begin{aligned} \langle {\delta }^{(k)},\phi \rangle = (-1)^k\phi ^{(k)}(0) \,. \end{aligned}$$

This example shows that, in general, to calculate the value $\langle T,\phi \rangle$ of a distribution T when applied to a test function $\phi$, it’s not enough to know the values of $\phi$ over ${\text {supp}(T)}$. We need to know the values of $\phi$ over a neighborhood of ${\text {supp}(T)}$.

Example 2.8: Derivative of $\boldsymbol{\textsf{1}_{+}}$

The derivative of the Heaviside unit step $\textsf{1}_{+}$ as a function is zero everywhere but at $t=0$ (a set of zero measure) where it is undefined. The function $\textsf{1}_{+}^{(1)}$ is therefore locally integrable, and we can define the regular distribution $T_{\textsf{1}_{+}^{(1)}}$ which evaluates to zero for every test function $\phi$.

Differently from this, the derivative of $\textsf{1}_{+}$ as a distribution is defined everywhere and, applying the definition, we find

$$\begin{aligned} \langle \textsf{1}_{+}^{(1)},\phi \rangle = -\langle \textsf{1}_{+},\phi ^{(1)} \rangle = -\int _0^\infty \,\phi ^{(1)}(\tau )\,d\tau = -\phi (\tau )|_0^\infty = \phi (0) = \langle {\delta },\phi \rangle \,, \end{aligned}$$

that is

$$\begin{aligned} \textsf{1}_{+}^{(1)} = {\delta }\,. \end{aligned}$$

Example 2.9: Function versus distributional derivative

Consider a function $f: {\mathbb {R}}\rightarrow {\mathbb {C}}$ continuously differentiable k times everywhere but at $t=0$, where it has a discontinuity such that both limits

$$\begin{aligned} \lim _{t\downarrow 0} f^{(i)}(t) \quad \text {and}\quad \lim _{t\uparrow 0} f^{(i)}(t) \end{aligned}$$

exist for all $i \le k$. Let’s denote the difference between these limits by

$$\begin{aligned} \alpha _i = \lim _{t\downarrow 0} f^{(i)}(t) - \lim _{t\uparrow 0} f^{(i)}(t) \,. \end{aligned}$$

Then we may represent the function f as

$$\begin{aligned} f(t) = f_{c,0}(t) + \alpha _0\textsf{1}_{+}(t) \end{aligned}$$

with $f_{c,0}$ a continuous function. It is easy to see that for $t\ne 0$, $f_{c,0}^{(1)}(t) = f^{(1)}(t)$. Thus, using the results of Example 2.8 the first order derivative of f is

$$\begin{aligned} T_f^{(1)} = T_{f^{(1)}} + \alpha _0{\delta }\,. \end{aligned}$$

To compute the second-order derivative we can use the same procedure. We decompose the function $f^{(1)}$ into a continuous function $f_{c,1}$ and a step

$$\begin{aligned} f^{(1)}(t) = f_{c,1}(t) + \alpha _1\textsf{1}_{+}(t) \,. \end{aligned}$$

Differentiating term by term we therefore obtain

$$\begin{aligned} T_f^{(2)} = & {} T_{f^{(1)}}^{(1)} + \alpha _0{\delta }^{(1)} \\ = & {} T_{f_{c,1}}^{(1)} + \alpha _1T_{\textsf{1}_{+}}^{(1)} + \alpha _0{\delta }^{(1)} \\ = & {} T_{f_{c,1}^{(1)}} + \alpha _1{\delta }+ \alpha _0{\delta }^{(1)} \\ = & {} T_{f^{(2)}} + \alpha _1{\delta }+ \alpha _0{\delta }^{(1)} \end{aligned}$$

The kth order derivative can be obtained by iterating this procedure

$$\begin{aligned} T_f^{(k)} = T_{f^{(k)}} + \alpha _0{\delta }^{(k-1)} + \alpha _1{\delta }^{(k-2)} + \cdots + \alpha _{k-1}{\delta }\,. \end{aligned}$$

(2.28)

Example 2.10: Logarithm derivative

In Example 2.6 we showed that the Cauchy principal value of $1/\tau$ is a distribution

$$\begin{aligned} \langle \textrm{pv}\,\frac{1}{\tau },\phi \rangle = & {} -\int _{-\infty }^{\infty }\ln |\tau |\,D\phi (\tau )\,d\tau . \end{aligned}$$

We now recognize this result as saying

$$\begin{aligned} \textrm{pv}\,\frac{1}{\tau } = & {} D\ln |\tau | \,. \end{aligned}$$

Example 2.11: Limit to $\boldsymbol{\infty }$ of trigonometric functions

Consider the following parameterized distribution

$$\begin{aligned} f_\omega (t) = -\frac{\cos \omega t}{\omega } \qquad \omega > 0 . \end{aligned}$$

As $\omega$ tends to infinity, it converges to

$$\begin{aligned} \lim _{\omega \rightarrow \infty } {|\langle f_\omega ,\phi \rangle |} {} & {} = \lim _{\omega \rightarrow \infty } \Bigg {|}{\int _{{\text {supp}(\phi )}} \, -\frac{\cos \omega t}{\omega }\, \phi (t)\, dt}\Bigg {|}\\ {} & {} \le \lim _{\omega \rightarrow \infty } \int _{{\text {supp}(\phi )}} \, \frac{|\phi (t)|}{\omega }\, dt\\ {} & {} \le \lim _{\omega \rightarrow \infty } \frac{\sup {|\phi (t)|}}{\omega } \, K \\ {} & {} = 0 \end{aligned}$$

with $K = {\text {supp}(\phi )}$. Since distributions can always be differentiated and for distributions arising from continuous functions the derivative as a distribution coincides with the derivative as a function, we have the following result

$$\begin{aligned} \lim _{\omega \rightarrow \infty } \langle \sin \omega t,\phi \rangle = & {} \lim _{\omega \rightarrow \infty } \langle -D\frac{\cos \omega t}{\omega },\phi \rangle \\ = & {} \lim _{\omega \rightarrow \infty } \langle \frac{\cos \omega t}{\omega },D\phi \rangle \\ = & {} 0 \end{aligned}$$

or

$$\begin{aligned} \lim _{\omega \rightarrow \infty } \sin \omega t = & {} 0 . \end{aligned}$$

(2.29)

Similarly one obtains

$$\begin{aligned} \lim _{\omega \rightarrow \infty } \cos \omega t = & {} 0 \quad \text {and}\end{aligned}$$

(2.30)

$$\begin{aligned} \lim _{\omega \rightarrow \infty } e^{\jmath \omega t} = & {} 0 . \end{aligned}$$

(2.31)

Note that these limits do not exist for the corresponding functions.

5 Distributions with Compact Support

The property of multiplication of a distribution with an indefinitely differentiable function suggests another interesting generalization. Let’s start again with a regular distribution f. Then, if we write out explicitly the integral of the distribution $\gamma \,f$

$$\begin{aligned} \langle \gamma \,f,\phi \rangle = & {} \int _{-\infty }^\infty f(\tau )\,\gamma (\tau )\,\phi (\tau )\,d\tau \end{aligned}$$

we see that in principle we could group the functions differently and write $\langle \phi \,f,\gamma \rangle$. This is however not a distribution in ${\mathcal {D'}}$ since $\gamma$ is not a test function as its support is not compact. In spite of this, the number $\langle \phi \,f,\gamma \rangle$ is the same as $\langle \gamma \,f,\phi \rangle$ for every test function $\phi$ and every function $\gamma \in {\mathcal {C}}^\infty$. A moment’s reflection reveals that what makes these two expressions have the same value for every value of $\gamma$ is the fact that, as a function, $\phi \,f$ has compact support.

To generalize this observation to arbitrary distributions we must first define what the support of a distribution T is.

A distribution is said to  vanish on an open set $U\in {\mathbb {R}}^n$ if $\langle T,\phi \rangle = 0$ for all test functions $\phi$ with ${\text {supp}(\phi )} \subset U$, where ${\text {supp}(\phi )}$ is the support of the test function $\phi$.

Definition 2.7

(Support of a distribution) The  support of a distribution T is the complement of the largest open set U on which the distribution vanishes and is denoted by ${\text {supp}(T)}$.

The set of all distributions with compact support is denoted by  ${\mathcal {E'}}$ and forms a vector subspace of ${\mathcal {D'}}$, that is ${\mathcal {E'}}\subset {\mathcal {D'}}$.

Example 2.12: Support of $\boldsymbol{{\delta }}$

The value of the Dirac delta distribution ${\delta }$ applied to any test function $\phi$ with ${\text {supp}(\phi )} \in U = (-\infty ,0)\cup (0,\infty )$ is zero. That is, ${\delta }$ vanishes on U. Its support is ${\text {supp}({\delta })} = {\mathbb {R}}\setminus U = \{0\}$ and is therefore compact.

With the notion of the support of a distribution we can generalize our observation that $\langle \phi \,f,\gamma \rangle = \langle \gamma \,f,\phi \rangle$ by saying that distributions with compact support $T\in {\mathcal {E'}}$ can be extended to continuous, linear functionals L on indefinitely differentiable functions with arbitrary support. In this context the vector space of all indefinitely differentiable functions is denoted by ${\mathcal {E}}$ and, to give a meaning to the continuity of the functionals, it is equipped with the following convergence criteria.

Definition 2.8

(Convergence in ${\mathcal {E}}$) A sequence $(\gamma _m)_{n\in {\mathbb {N}}} \in {\mathcal {E}}$ is said to converge to $\gamma$ if for every compact subset K of ${\mathbb {R}}^n$ and every n-tuple k, the set of functions $D^k\gamma _m$ converges uniformly to $D^k\gamma$

$$\begin{aligned} \sup _{x\in K} \left| D^k\gamma _m - D^k\gamma \right| \rightarrow 0, \qquad m \rightarrow \infty . \end{aligned}$$

Assume that the support of T is the compact set K and let $\alpha$ be a test function equal to 1 in a neighborhood U of K. Then for every function $\gamma \in {\mathcal {E}}$ and for every point $\tau \in U$, $\alpha (\tau )\,\gamma (\tau ) = \gamma (\tau )$. Therefore, there is a functional L such that

$$\begin{aligned} \langle L,\gamma \rangle = \langle T,\alpha \gamma \rangle \qquad \gamma \in {\mathcal {E}} . \end{aligned}$$

(2.32)

That it is independent of the choice of $\alpha$ is easily verified: Suppose that $\alpha _1$ and $\alpha _2$ are two test functions equal to 1 in a neighborhood of K. Then in the smallest of these neighborhoods $\alpha _1 - \alpha _2 = 0$ and $\langle T,\alpha _1\,\gamma \rangle = \langle T,\alpha _2\,\gamma \rangle$.

The functional thus defined is unique since, for every sequence of test functions $\alpha _m$ equal to 1 for $|\tau | < m$ we have: on the one hand, by continuity of L

$$\begin{aligned} \lim _{m\rightarrow \infty } \langle L,\alpha _m \, \gamma \rangle = \langle L,\gamma \rangle \end{aligned}$$

and on the other hand, for sufficiently large m

$$\begin{aligned} \langle T,\alpha _m \, \gamma \rangle = \langle L,\gamma \rangle \,. \end{aligned}$$

Therefore every distribution with compact support T defines a unique continuous, linear functional L on ${\mathcal {E}}$.

The converse is also true: Every continuous, linear functional L restricted to ${\mathcal {D}} \subset {\mathcal {E}}$ defines a distribution T with compact support. For, if this was not the case and the support of T was not compact, then we could find a sequence of test functions $\phi _m \in {\mathcal {D}}$ with support in the complement of $|\tau | < m$, such that $\langle T,\phi _m \rangle = 1$ for all m. However, since in ${\mathcal {E}}$ $\lim _{m\rightarrow \infty } \phi _m = 0$, by continuity of L

$$\begin{aligned} \lim _{m\rightarrow \infty } \langle L,\phi _m \rangle = 0\,. \end{aligned}$$

Therefore if $\langle L,\phi \rangle = \langle T,\phi \rangle$ for all $\phi \in {\mathcal {D}}$, then the support of T must be compact.

There are other vector sub-spaces of ${\mathcal {D'}}$ which can be extended to larger sets of functions than ${\mathcal {D}}$. We will encounter another one in the context of the Fourier transform. The set of test functions ${\mathcal {D}}$ are the common set on which all distributions are defined.

5.1 Single-Point Support

We now investigate distributions satisfying the following equation

$$\begin{aligned} t^k \, T = 0\,, \end{aligned}$$

(2.33)

that is, distributions for which, for every $k \ge 1$ and test function $\phi$

$$\begin{aligned} \langle t^k \, T,\phi \rangle = \langle T,t^k \, \phi \rangle = 0\,. \end{aligned}$$

For simplicity we limit ourselves to the one dimensional case.

First observe that on the open set $U = (-\infty ,0) \cup (0,\infty )$ the function $t \mapsto t^k$ doesn’t assume the zero value. For this reason, to satisfy the equation, T must vanish on U, or, stated in other words, the support of T must be the origin: ${\text {supp}(T)} = \{0\}$.

Since the support of T is compact (a single point) and, for any test function $\phi$, the value of $\langle T,\phi \rangle$ is determined by the values of $\phi$ in a neighborhood (however small) of ${\text {supp}(T)}$, we can expand $\phi$ using Taylor’s formula with remainder [19]. For our purposes it is convenient to express the remainder in integral form which can be obtained by integrating by parts multiple times

$$\begin{aligned} \phi (t) = & {} \phi (0) + \int _0^t\, \phi ^{(1)}(\tau ) \, d\tau \\ = & {} \phi (0) - \left. (t - \tau ) \phi ^{(1)}(\tau ) \right| _0^t \, + \int _0^t\, (t - \tau ) \, \phi ^{(2)}(\tau ) \, d\tau \\ = & {} \phi (0) + t\,\phi ^{(1)}(0) - \left. \frac{(t - \tau )^2}{2} \phi ^{(2)}(\tau ) \right| _0^t \, + \int _0^t\, \frac{(t - \tau )^2}{2} \, \phi ^{(3)}(\tau ) \, d\tau \\ = & {} \cdots \\ = & {} \sum _{m=0}^{k-1} \, \frac{\phi ^{(m)}(0)}{m!} \, t^m + \int _0^t\, \frac{(t-\tau )^{k-1}}{(k-1)!} \, \phi ^{(k)}(\tau ) \, d\tau . \end{aligned}$$

By performing the substitution $\tau = t\,\lambda$ the remainder can be transformed in the following form

$$\begin{aligned} t^k \, \int _0^1\, \frac{(1-\lambda )^{k-1}}{(k-1)!} \, \phi ^{(k)}(t\,\lambda ) \, d\lambda = & {} \frac{t^k}{(k-1)!} \, \psi (t) \end{aligned}$$

which makes it apparent that it is proportional to the product of $t^k$ and an indefinitely differentiable function $\psi \in {\mathcal {E}}$. Note that, differently from $\phi$, no addend has compact support. This poses no problem since T, having itself compact support, can be extended uniquely to a distribution on ${\mathcal {E}}$ (see Sect. 2.5).

With this expansion we can express the value of $\langle T,\phi \rangle$ as a finite sum. Taking into account (2.33)

$$\begin{aligned} \langle T,\phi \rangle = & {} \sum _{m=0}^{k-1} \, \frac{\phi ^{(m)}(0)}{m!} \, \langle T,t^m \rangle + \frac{1}{(k-1)!} \langle T,t^k \, \psi (t) \rangle \\ = & {} \sum _{m=0}^{k-1} \, \frac{\phi ^{(m)}(0)}{m!} \, \langle T,t^m \rangle \\ = & {} \sum _{m=0}^{k-1} \, c_m \, \langle {\delta }^{(m)},\phi \rangle \end{aligned}$$

or

$$\begin{aligned} T = \sum _{m=0}^{k-1} c_m {\delta }^{(m)} \end{aligned}$$

(2.34)

with

$$\begin{aligned} c_m = & {} (-1)^m \, \frac{\langle T,t^m \rangle }{m!}\,. \end{aligned}$$

We have therefore established that the homogeneous equation

$$\begin{aligned} t^k \, T = 0 \end{aligned}$$

has an infinity of non-trivial solutions, each being a weighted sum of the Dirac delta distribution ${\delta }$ and its derivatives up to order $k-1$. In addition, this shows that ${\delta }$ and its derivatives are the only distributions with the support consisting of a single point.

Example 2.13: Solutions of $\boldsymbol{t\,T=1}$

We want to find all solutions of the equation

$$\begin{aligned} t \, T = 1\,. \end{aligned}$$

If T would be a function then the equation would have no solution at $t=0$ and 1/t at all other points. From this we guess that the solution as a distribution could be $T=\textrm{pv}\, 1/t$. Indeed, this distribution satisfies the equation

$$\begin{aligned} \Big \langle t\,\textrm{pv}\frac{1}{t},\phi \Big \rangle = & {} \Big \langle \textrm{pv}\frac{1}{t},t\,\phi \Big \rangle \\ = & {} \lim _{\epsilon \downarrow 0} \int _{t \ge \epsilon }\, \frac{1}{t}\,t\,\phi (t)\, dt \\ = & {} \int _{-\infty }^{\infty } \phi (t) \, dt \\ = & {} \langle 1,\phi \rangle \,. \end{aligned}$$

However, this is not the only solution as the homogeneous equation has non-trivial solutions given by (2.34) (k $=$ 1). The equation is therefore satisfied by all distributions of the form

$$\begin{aligned} T = \textrm{pv}\frac{1}{t} + c \, {\delta }(t) \end{aligned}$$

with c an arbitrary constant.

The properties of distributions discussed in this chapter and some that will be discussed in the following ones are summarised in Table 2.1.

Table 2.1 Main properties and operations on distributions