Fourier Transform of Distributions

Abstract

The Fourier transform is a major tool in the analysis of signals and systems. We will see that its extension to distributions will make the derivation of many results simpler and more direct than when working with functions.

The Fourier transform is a major tool in the analysis of signals and systems. We will see that its extension to distributions will make the derivation of many results simpler and more direct than when working with functions.

1 Test Functions of Fast Descent

Consider a Lebesgue integrable function f. Its Fourier transform is defined by

$$\begin{aligned} {\mathcal {F}}\{f\}(\omega ) :=\hat{f}(\omega ) :=\int \limits _{-\infty }^\infty f(t) \, \text {e}^{-\jmath \omega t} \, dt \end{aligned}$$

(4.1)

which is a continuous function of $\omega$. If f is such that $\hat{f}$ is also integrable, then

$$\begin{aligned} f(t) = {\mathcal {F}}^{-1}\{\hat{f}\}(t) :=\frac{1}{2\pi } \int \limits _{-\infty }^\infty \hat{f}(\omega ) \, \text {e}^{\jmath \omega t} \, d\omega \end{aligned}$$

(4.2)

almost everywhere, with ${\mathcal {F}}^{-1}\{\hat{f}\}$ the inverse Fourier transform. ${\mathcal {F}}^{-1}\{\hat{f}\}$ may differ from f at the points where f is not continuous.

The Fourier transform of the regular distribution f is thus

$$\begin{aligned} \left\langle \hat{f}(\omega ),\phi (\omega ) \right\rangle {} & {} = \int \limits _{-\infty }^\infty \int \limits _{-\infty }^\infty f(t) \, \text {e}^{-\jmath \omega t} \, dt \, \phi (\omega ) \, d\omega \nonumber \\ {} & {} = \int \limits _{-\infty }^\infty f(t) \int \limits _{-\infty }^\infty \phi (\omega ) \, \text {e}^{-\jmath \omega t} \, d\omega \, dt \nonumber \\ {} & {} = \int \limits _{-\infty }^\infty f(t) \hat{\phi }(t) \, dt\,. \end{aligned}$$

(4.3)

The last integral looks like a distribution in a form suitable to be generalized to arbitrary distributions. However, the support of $\hat{\phi }$ is not compact. This is a manifestation of the uncertainty principle of the Fourier transform and can readily be seen by

$$\begin{aligned} \hat{\phi }(\omega ) {} & {} = \int \limits _{-\infty }^\infty \phi (t) \, \text {e}^{-\jmath \omega t} \, dt = \int \limits _{-a}^b \sum _{m=0}^\infty \frac{(-\jmath \omega )^m}{m!} \, t^m \, \phi (t) \, dt \nonumber \\ {} & {} = \sum _{m=0}^\infty \frac{(-\jmath \omega )^m}{m!} \, \int \limits _{-a}^b \phi (t) \, t^m \, dt \end{aligned}$$

(4.4)

with a and b constants such that the interval [a, b] includes the support of $\phi$.

To obtain a definition of the Fourier transform suitable for arbitrary distributions we have to replace the space of test functions ${\mathcal {D}}$ with a space closed under Fourier transformation. Suitable characteristics for the functions in this space can be inferred from the above expression for $\hat{\phi }$. First, given the uncertainty principle, the space has to be extended to functions of unbounded support (and therefore, the last step, the exchange of summation and integration, may not be valid). Then, if all summands have to remain finite, the limits $\lim _{t \rightarrow \pm \infty } \phi (t)\, t^m$ have to converge to zero for all values of m. Finally, to preserve arbitrary differentiability, the above characteristics must be satisfied by all derivatives of $\phi$. These are the characteristics of the so-called Schwartz space  ${\mathcal {S}}$ of which we give the general definition.

Definition 4.1

(Schwartz space ${\mathcal {S}}({\mathbb {R}}^n)$) The Schwartz space ${\mathcal {S}}({\mathbb {R}}^n)$ is the vector space of indefinitely differentiable functions $\phi : {\mathbb {R}}^n \rightarrow {\mathbb {C}}$ that, together with all their derivatives, decrease more rapidly than any power of $1/|\tau |$ as $|\tau | \rightarrow \pm \infty$. That is, for any n-tuples $m,k \in {\mathbb {N}}^n$ and $\tau \in {\mathbb {R}}^n$

$$\begin{aligned} \lim _{|\tau | \rightarrow \pm \infty } |\tau ^m \, D^k\phi (\tau )| = 0\,. \end{aligned}$$

(4.5)

Functions $\phi$ in the Schwartz space are called test functions of rapid descent, or Schwartz functions.

To see that the Fourier transform of a function $\phi \in {\mathcal {S({\mathbb {R}})}}$ is indeed another function in the same space, consider the kth derivative of $\hat{\phi }$. By integrating by parts we find

$$\begin{aligned} D^k\hat{\phi }(\omega ) {} & {} = \int \limits _{-\infty }^\infty (-\jmath t)^k \, \phi (t) \, \text {e}^{-\jmath \omega t} \, dt \\ {} & {} = \frac{1}{\jmath \omega } \int \limits _{-\infty }^\infty \text {e}^{-\jmath \omega t} D\left[ (-\jmath t)^k \, \phi (t) \right] \, dt \end{aligned}$$

and by iterating m times

$$\begin{aligned} \left| (\jmath \omega )^m \, D^k\hat{\phi }(\omega ) \right| {} & {} = \left| \int \limits _{-\infty }^\infty \text {e}^{-\jmath \omega t} D^m\left[ (-\jmath t)^k \, \phi (t) \right] \, dt \right| \\ {} & {} \le \int \limits _{-\infty }^\infty \left| D^m\left[ t^k \, \phi (t) \right] \right| \, dt\,. \end{aligned}$$

Since this is valid for arbitrary k and m it shows that $\hat{\phi }$ is in fact a function in the Schwartz space. In addition, given that the Fourier transform, and its inverse are almost symmetric, a similar calculation shows that the inverse Fourier transform of a Schwartz function $\hat{\phi }$ is a function $\phi \in {\mathcal {S}}$. That is, the Fourier transform is a bijection from the space ${\mathcal {S}}$ into itself.

Example 4.1: Gauss Function

An important example of a function of rapid descent is the Gauss function

$$\begin{aligned} \phi (t) = \frac{1}{\sqrt{2\pi }\sigma } \text {e}^{-t^2/(2\sigma ^2)}\,. \end{aligned}$$

It’s widely known that its Fourier transform is

$$\begin{aligned} \hat{\phi }(\omega ) = \text {e}^{-\omega ^2\sigma ^2/2}\,. \end{aligned}$$

One of the defining characteristics of distributions is their continuity. To talk about continuity we introduce a convergence principle (topology) similar to the ones we defined for ${\mathcal {D}}$ and ${\mathcal {E}}$.

Definition 4.2

(Convergence in ${\mathcal {S}}({\mathbb {R}}^n)$) A sequence of functions $\phi _m \in {\mathcal {S}}({\mathbb {R}}^n)$ is said to converge in ${\mathcal {S}}({\mathbb {R}}^n)$ to a function $\phi \in {\mathcal {S}}({\mathbb {R}}^n)$, if for each n-tuples $k,p \in {\mathbb {N}}^n$ and $\tau \in {\mathbb {R}}^n$ the sequence $|\tau |^p \, D^k\phi _m(\tau )$ converges uniformly to $|\tau |^p \, D^k\phi (\tau )$, that is if

$$\begin{aligned} \lim _{m \rightarrow \infty } |\tau |^p \, D^k\phi _m(\tau ) = |\tau |^p \, D^k\phi (\tau )\,. \end{aligned}$$

2 Fourier Transform of Tempered Distributions

For (4.3) to be an expression suitable for the definition of the Fourier transform for an arbitrary distribution, we must verify its linearity and continuity. The former is clear. To show the latter we have to verify that, if a sequence of test functions $\phi _m \in {\mathcal {S}}$ converges to zero, so does the sequence of their Fourier transforms $\hat{\phi }_m$. That this is the case is shown by the following upper bound

$$\begin{aligned} |\hat{\phi }_m(\omega ) | {} & {} = \bigg {|}{\int \limits _{-\infty }^\infty \phi _m(t) \, \text {e}^{-\jmath \omega t} \, dt} \le \int _{|t| < 1} |\phi _m(t)| \, dt + \int _{|t| \ge 1} |\phi _m(t)| \, dt \\ {} & {} \le 2 \sup _{|t| < 1}{|\phi _m(t)|} + \int _{|t| \ge 1} |\frac{\phi _m(t) \, t^2}{t^2}| \, dt \\ {} & {} \le 2 \sup _{|t| < 1}{|\phi _m(t)|} + \sup _{|t| \ge 1}{|\phi _m(t) \, t^2|} \int _{|t| \ge 1} \frac{1}{|t^2|} \, dt \\ {} & {} = 2 \sup _{|t| < 1}{|\phi _m(t)|} + 2 \sup _{|t| \ge 1}{|\phi _m(t) \, t^2|}\,. \end{aligned}$$

We have a good candidate for the definition of the Fourier transform for an arbitrary distribution. However, since the space ${\mathcal {S}}$ is larger than ${\mathcal {D}}$, the Fourier transform can only be defined for the following subset  of distributions.

Definition 4.3

(Tempered distributions) Tempered distributions (also called distributions of slow growth) are distributions that can be extended to continuous, linear functionals on the Schwartz space ${\mathcal {S}}$.

The space of all continuous, linear functionals on ${\mathcal {S}}$ is denoted by ${\mathcal {S'}}$ and, since the Schwartz space ${\mathcal {S}}$ is a subspace of ${\mathcal {E}}$, we have the following inclusion: ${\mathcal {E'}} \subset {\mathcal {S'}} \subset {\mathcal {D'}}$. Consequently, from Sect. 2.5, we conclude that, if a distribution $T \in {\mathcal {D'}}$ can be extended to a continuous, linear functional on ${\mathcal {S}}$, then this extension is unique (and the other way around). ${\mathcal {S'}}$ can therefore be identified with tempered distributions.

Example 4.2: Slowly Increasing Function

Consider a locally integrable function f satisfying

$$\begin{aligned} |f(t)| \le C \, |t|^m \qquad \text {as} \quad |t| \rightarrow \infty \end{aligned}$$

for some constant C and some natural number m. Then, f is a tempered distribution, since

$$\begin{aligned} |\left\langle f,\phi \right\rangle | &\le & \int _{|t| < 1} \left| f(t) \, \phi (t) \right| dt + \int _{|t| \ge 1} \left| f(t) \, \phi (t) \right| dt\\ &\le & \sup _{|t| < 1} |\phi (t)| \int _{|t| < 1} | f(t) | dt + \sup _{|t| \ge 1} \left( |t^{m+2}| \, |\phi (t)| \right) \int _{|t| \ge 1} \frac{C}{|t|^2} dt \end{aligned}$$

is bounded for every $\phi \in {\mathcal {S}}$.

Example 4.3: Distributions in $\boldsymbol{{\mathcal {E'}}}$

Distribution with bounded support are defined on all indefinitely differentiable function, independently of their asymptotic behavior as $t \rightarrow \infty$. For this reason the Fourier transform of distributions in ${\mathcal {E'}}$ is always well-defined.

Example 4.4: Multiplication with Polynomial

If T is a tempered distribution and p a polynomial, then $p \, T$ is a tempered distribution. $p \, T$ is in fact defined as

$$\begin{aligned} \left\langle p \, T,\phi \right\rangle = \left\langle T,p \, \phi \right\rangle \end{aligned}$$

and it’s easy to see that $p \, \phi \in {\mathcal {S}}$.

We can finally define the Fourier transform for tempered distributions.

Definition 4.4

(Fourier transform on ${\mathcal {S'}}$) The Fourier transform of a tempered distribution T and its inverse, are defined by

$$\begin{aligned} \left\langle {\mathcal {F}}\{T\},\phi \right\rangle := & {} \left\langle T,{\mathcal {F}}{\{\phi \}} \right\rangle \end{aligned}$$

(4.6)

$$\begin{aligned} \left\langle {\mathcal {F}}^{-1}\{T\},\phi \right\rangle := & {} \left\langle T,{\mathcal {F}}^{-1}{\{\phi \}} \right\rangle \end{aligned}$$

(4.7)

for every function $\phi \in {\mathcal {S}}$.

Clearly, the Fourier transform of a tempered distribution is a tempered distribution. Note that, given the properties of Schwartz functions, for a tempered distribution it’s always the case that

$$\begin{aligned} {\mathcal {F}}^{-1}\{{\mathcal {F}}\left\{ T\right\} \} = {\mathcal {F}}\left\{ {\mathcal {F}}^{-1}\{T\}\right\} = T\,. \end{aligned}$$

In addition the Fourier transform and its inverse satisfy the following symmetry relation

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ T\right\} ,\phi \right\rangle {} & {} = \left\langle T,{\mathcal {F}}\left\{ \phi \right\} \right\rangle = \left\langle T(\omega ),\int \limits _{-\infty }^\infty \phi (t) \text {e}^{-\jmath \omega t} dt \right\rangle \\ {} & {} = \left\langle T(\omega ),2\pi {\mathcal {F}}^{-1}\{\phi \}(-\omega ) \right\rangle \\ {} & {} = \left\langle 2\pi \, {\mathcal {F}}^{-1}\{T(-\omega )\},\phi \right\rangle \,. \end{aligned}$$

If in this expression we replace T by its Fourier transform and denote it by $\hat{T}$, then this symmetry relation can also be expressed as

$$\begin{aligned} {\mathcal {F}}\left\{ \hat{T}(t)\right\} = 2\pi \, T(-\omega )\,. \end{aligned}$$

(4.8)

As with functions, we will often use the convention of denoting by $\hat{T}$ the Fourier transform of a tempered distribution T.

Example 4.5: Fourier Transform and $\boldsymbol{\delta }$

The Fourier transform of the delta distribution $\delta$ is

$$\begin{aligned} \left\langle \hat{\delta },\phi \right\rangle = \left\langle \delta ,\hat{\phi } \right\rangle = \hat{\phi }(0) = \int \limits _{-\infty }^\infty \phi (t) dt = \left\langle 1,\phi \right\rangle \end{aligned}$$

or

$$\begin{aligned} \hat{\delta } = 1\,. \end{aligned}$$

Conversely, the Fourier transform of the constant function 1 is

$$\begin{aligned} \left\langle \hat{1},\phi \right\rangle = \left\langle 1,\hat{\phi } \right\rangle = \int \limits _{-\infty }^\infty \hat{\phi }(\omega ) \, d\omega = 2\pi \, \left\langle \delta ,{\mathcal {F}}^{-1}\{\hat{\phi }\} \right\rangle = 2\pi \, \left\langle \delta ,\phi \right\rangle \end{aligned}$$

or

$$\begin{aligned} \hat{1} = 2\pi \, \delta \,. \end{aligned}$$

This expression is often found in the technical literature symbolically written as

$$\begin{aligned} \delta (t) = \frac{1}{2\pi } \int \limits _{-\infty }^\infty \text {e}^{-\jmath \omega t} \, d\omega = \frac{1}{2\pi } \int \limits _{-\infty }^\infty \text {e}^{\jmath \omega t} \, d\omega \,. \end{aligned}$$

The Fourier transform of the derivative of $\delta$ is

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ D\delta \right\} ,\phi \right\rangle = \left\langle D\delta ,\hat{\phi } \right\rangle = -\left\langle \delta ,-\jmath \omega \hat{\phi } \right\rangle = \jmath \omega \left\langle \hat{\delta },\phi \right\rangle = \left\langle \jmath \omega ,\phi \right\rangle \end{aligned}$$

and, by iterating this procedure, for the higher order derivatives we find

$$\begin{aligned} {\mathcal {F}}\left\{ D^k\delta \right\} = (\jmath \omega )^k\,. \end{aligned}$$

Example 4.6: Complex Tones

The Fourier transform of a complex tone is

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ \text {e}^{\jmath \omega _c t}\right\} ,\phi \right\rangle {} & {} = \left\langle \text {e}^{\jmath \omega _c t},\hat{\phi } \right\rangle = \int \limits _{-\infty }^{\infty } \text {e}^{\jmath \omega _c t} \hat{\phi }(t) \, dt = 2\pi \, \phi (\omega _c) \\ {} & {} = 2\pi \, \left\langle \delta (\omega - \omega _c),\phi \right\rangle \end{aligned}$$

or

$$\begin{aligned} {\mathcal {F}}\left\{ \text {e}^{\jmath \omega _c t}\right\} = 2\pi \, \delta (\omega - \omega _c)\,. \end{aligned}$$

Similarly, the Fourier transform of a shifted Dirac pulse is found to be

$$\begin{aligned} {\mathcal {F}}\left\{ \delta (t - \tau _o)\right\} = \text {e}^{-\jmath \omega \tau _0}\,. \end{aligned}$$

Example 4.7: Dirac comb

An equally spaced sequence of Dirac pulses is a tempered distribution called a  Dirac comb with period ${\mathcal {T}}$

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

The linearity and continuity of distributions permit to calculate its Fourier transform term by term and, by using previous results, we obtain

$$\begin{aligned} {\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\} = \sum _{m=-\infty }^\infty \text {e}^{\jmath \omega m {\mathcal {T}}}\,. \end{aligned}$$

This distribution is formally the limit

$$\begin{aligned} \lim _{K,M \rightarrow \infty } \left\langle s_{P,K}(\omega ) + s_{N,M}(\omega ) - 1,\phi (\omega ) \right\rangle \end{aligned}$$

with

$$\begin{aligned} s_{P,K}(\omega ) := & {} \sum _{m=0}^{K-1} \text {e}^{\jmath \omega m {\mathcal {T}}} \\ s_{N,M}(\omega ) := & {} \sum _{m=0}^{M-1} \text {e}^{-\jmath \omega m {\mathcal {T}}}. \end{aligned}$$

For values of $\omega \ne k \, 2\pi /{\mathcal {T}}, k\in {\mathbb {Z}}$ the partial sums can be represented by

$$\begin{aligned} s_{P,K}(\omega ) {} & {} = \frac{1 - \text {e}^{\jmath \omega K {\mathcal {T}}}}{1 - \text {e}^{\jmath \omega {\mathcal {T}}}} = \frac{1}{1 - \text {e}^{\jmath \omega {\mathcal {T}}}} - \frac{\text {e}^{\jmath \omega K {\mathcal {T}}}}{1 - \text {e}^{\jmath \omega {\mathcal {T}}}} \\ s_{N,M}(\omega ) {} & {} = \frac{1 - \text {e}^{-\jmath \omega M {\mathcal {T}}}}{1 - \text {e}^{-\jmath \omega {\mathcal {T}}}} = \frac{1}{1 - \text {e}^{-\jmath \omega {\mathcal {T}}}} - \frac{\text {e}^{-\jmath \omega M {\mathcal {T}}}}{1 - \text {e}^{-\jmath \omega {\mathcal {T}}}}\,. \end{aligned}$$

The sum of the first terms is easily seen to equal 1 and, with the results of Example 2.11, the limit of the second ones do vanish. The support of ${\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\}$ therefore consists in the set of points $\omega = k \, 2\pi /{\mathcal {T}}, k\in {\mathbb {Z}}$. Consequently, when applied to any test function $\phi \in {\mathcal {S}}$, its value must be a weighted sum of the values of the test function at these points. Since, replacing $\phi (t)$ by $\phi (t+{\mathcal {T}})$ doesn’t change the result, we can also deduce that the weighting factor must be the same for all terms. We thus have

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\} ,\phi \right\rangle = \sum _{m=-\infty }^\infty C \phi (m\omega _c) = C \left\langle \delta _{\omega _c},\phi \right\rangle \end{aligned}$$

with C a constant and $\omega _c=2\pi /{\mathcal {T}}$. The value of the constant can be found by inserting any Schwartz function. A convenient choice is the one of Example 4.1 with $\sigma =\sqrt{2\pi }/T$. With it, on the one hand we have

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\} ,\phi \right\rangle = C \left\langle {\delta _{\omega _c}} {\frac{{\mathcal {T}}}{2\pi } \text {e}^{-(t T)^2/(4\pi )}},= \right\rangle C \frac{{\mathcal {T}}}{2\pi } \sum _{m=-\infty }^\infty \text {e}^{-m^2\pi } \end{aligned}$$

and on the other hand

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\} ,\phi \right\rangle = \left\langle \delta _{\mathcal {T}},\hat{\phi } \right\rangle = \left\langle \delta _{\mathcal {T}},\text {e}^{-(t/T)^2 \pi } \right\rangle = \sum _{m=-\infty }^\infty \text {e}^{-m^2\pi } \end{aligned}$$

so that $C=2\pi /T$. We have thus established the following important result

$$\begin{aligned} {\mathcal {F}}\left\{ \delta _{\mathcal {T}}\right\} = \omega _c \, \delta _{\omega _c}\,. \end{aligned}$$

(4.9)

The Fourier transforms of the $\delta$ and related distributions are summarized in Table 4.1.

Table 4.1 Fourier transformation of the $\delta$ and related distributions

A useful property of the Fourier transform is that it preserves parity. This means that the Fourier transform of an odd tempered distribution T is odd

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ T\right\} ,\phi (-t) \right\rangle {} & {} = \left\langle T,{\mathcal {F}}\left\{ \phi (-t)\right\} \right\rangle = \left\langle T,\int _{\mathbb {R}}\phi (-t) \, \text {e}^{-\jmath \omega t} dt \right\rangle \\ {} & {} = \left\langle T,\int _{\mathbb {R}}\phi (t) \, \text {e}^{\jmath \omega t} dt \right\rangle \\ {} & {} = \left\langle T,\hat{\phi }(-\omega ) \right\rangle = - \left\langle T,\hat{\phi }(\omega ) \right\rangle \\ {} & {} = - \left\langle {\mathcal {F}}\left\{ T\right\} ,\phi (t) \right\rangle \end{aligned}$$

and, similarly, the Fourier transform of an even tempered distribution is even.

We conclude this section with an important property of the Fourier transform of real distributions. Let T be a real distribution, $\phi$ a real valued Schwartz function and let denote complex conjugation by an over bar. Then

$$\begin{aligned} \begin{aligned} \left\langle \overline{\hat{T}},\phi \right\rangle &= \overline{\left\langle \hat{T},\phi \right\rangle } = \overline{\left\langle T,\hat{\phi } \right\rangle } = \left\langle T,\overline{\hat{\phi }} \right\rangle \\ &= \left\langle T,\hat{\phi }(-\omega ) \right\rangle = \left\langle \hat{T}(-\omega ),\phi \right\rangle \end{aligned} \end{aligned}$$

or

$$\begin{aligned} \overline{\hat{T}}(\omega ) = \hat{T}(-\omega ) \end{aligned}$$

(4.10)

as for real functions.

3 Distributions with Bounded Support

The Fourier transform of distributions with bounded support can be expressed in a simpler, useful form that we explore in this section. To this end, consider first the convolution between a distribution of bounded support T and the regular constant distribution 1

$$\begin{aligned} \left\langle 1 *T,\phi \right\rangle {} & {} = \left\langle T,\left\langle 1,\phi \right\rangle \right\rangle = \left\langle T(\tau ),\int _{\mathbb {R}}\phi (\tau + \lambda ) \, d\lambda \right\rangle \\ {} & {} = \left\langle 1,\left\langle T,\phi \right\rangle \right\rangle = \int _{\mathbb {R}}\left\langle T(\tau ),\phi (\tau + \lambda ) \right\rangle \, d\lambda \end{aligned}$$

and note the two equivalent integral representations. With these equalities we can then proceed to represent the Fourier transform of T by

$$\begin{aligned} \left\langle \hat{T},\phi \right\rangle {} & {} = \left\langle T(t),\int _{\mathbb {R}}\phi (\omega ) \text {e}^{-\jmath t \omega } \, d\omega \right\rangle \\ {} & {} = \int _{\mathbb {R}}\left\langle T(t),\phi (\omega ) \text {e}^{-\jmath t \omega } \right\rangle d\omega \\ {} & {} = \int _{\mathbb {R}}\left\langle T(t),\text {e}^{-\jmath \omega t} \right\rangle \phi (\omega ) \, d\omega \\ {} & {} = \left\langle \left\langle T(t),\text {e}^{-\jmath \omega t} \right\rangle ,\phi (\omega ) \right\rangle \end{aligned}$$

with

$$\begin{aligned} \hat{T}(\omega ) = \left\langle T(t),\text {e}^{-\jmath \omega t} \right\rangle \end{aligned}$$

(4.11)

an indefinitely differentiable function of slow growth. In a similar way we obtain

$$\begin{aligned} {\mathcal {F}}^{-1}\{T\} = \frac{1}{2\pi }\left\langle T(\omega ),\text {e}^{\jmath \omega t} \right\rangle \,. \end{aligned}$$

(4.12)

4 Fourier Transform and Convolution

Consider a tempered distribution S and a distribution with bounded support T. Their convolution is well-defined if, for every test function $\phi \in {\mathcal {S}}$

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

In the first case,

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

is a function $\zeta (\tau ) \in {\mathcal {S}}$ and the outer functional is therefore well-defined. In the second case,

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

is an indefinitely differentiable function $\gamma (\tau ) \in {\mathcal {E}}$. Consequently, the outer functional is again well-defined. Equality of the two expressions is guaranteed by the uniqueness of the extension of the distributions T and S to ${\mathcal {E'}}$ and ${\mathcal {S'}}$ respectively (and the other way around).

We now show that the Fourier transform of the convolution of S and T is the product of their Fourier transforms $\hat{S}$ and $\hat{T}$:

$$\begin{aligned} \left\langle {\mathcal {F}}\left\{ S *T\right\} ,\phi \right\rangle {} & {} = \left\langle S *T,{\mathcal {F}}\left\{ \phi \right\} \right\rangle = \left\langle S,\left\langle T,{\mathcal {F}}\left\{ \phi \right\} \right\rangle \right\rangle \\ {} & {} = \left\langle S(\omega ),\left\langle T(\lambda ),\int _{\mathbb {R}}\phi (t) \text {e}^{-\jmath (\omega + \lambda ) t} \, dt \right\rangle \right\rangle \\ {} & {} = \left\langle S(\omega ),\int _{\mathbb {R}}\phi (t) \left\langle T(\lambda ),\text {e}^{-\jmath \lambda t} \right\rangle \text {e}^{-\jmath \omega t} \, dt \right\rangle \\ {} & {} = \left\langle {\mathcal {F}}\left\{ S(\omega )\right\} ,\phi (t) \left\langle T(\lambda ),\text {e}^{-\jmath \lambda t} \right\rangle \right\rangle \\ {} & {} = \left\langle {\mathcal {F}}\left\{ S(\omega )\right\} \left\langle T(\lambda ),\text {e}^{-\jmath \lambda t} \right\rangle ,\phi (t) \right\rangle \\ {} & {} = \left\langle \hat{S} \, \hat{T},\phi \right\rangle \end{aligned}$$

or

$$\begin{aligned} {\mathcal {F}}\left\{ S *T\right\} = \hat{S} \, \hat{T}\,. \end{aligned}$$

(4.13)

The product is well-defined since $\hat{T}$ is an indefinitely differentiable function of slow growth. A similar result is readily obtained for the inverse Fourier transform

$$\begin{aligned} {\mathcal {F}}^{-1}\{S *T\} = 2\pi \, {\mathcal {F}}^{-1}\{S\} {\mathcal {F}}^{-1}\{T\}\,. \end{aligned}$$

(4.14)

These are central results and arguably the most important properties of the Fourier transformation. It can be shown that this relation is valid in other cases as well. For example, in the case of locally integrable functions which are slowly increasing [20].

With these properties, the previously obtained Fourier transforms for the Dirac $\delta$ distribution and the properties of the convolution product we immediately obtain the properties listed in Table 4.2. In particular, it’s noteworthy the fact that the Dirac $\delta$ distribution acting as a unit with respect to the convolution product is related to the fact that its Fourier transform is 1.

Table 4.2 Properties of the Fourier transformation

Example 4.8: Fourier Transform of $\boldsymbol{\textrm{pv} \, 1/t}$

In Example 2.13we saw that the equation

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

has solutions

$$\begin{aligned} T = \textrm{pv} \, \frac{1}{t} + C\delta . \end{aligned}$$

with C a constant. By noting that $\textrm{pv} \, 1/t$ is odd, while $\delta$ is even, we can find the Fourier transform of the former. First observe that

$$\begin{aligned} {\mathcal {F}}\left\{ (-\jmath t) \, T\right\} = D\hat{T} \end{aligned}$$

and hence, by transforming both sides of the equation we have that

$$\begin{aligned} \jmath \, D\hat{T} = 2\pi \, \delta \,. \end{aligned}$$

Since the Fourier transform preserves parity, we have to look for an odd solution of this equation, and we find

$$\begin{aligned} \hat{T}(\omega ) = -\jmath \pi \, \textrm{sign}(\omega ) \qquad \text {or} \qquad {\mathcal {F}}\left\{ \textrm{pv} \, \frac{\jmath }{\pi \, t}\right\} (\omega ) = \textrm{sign}(\omega )\,. \end{aligned}$$

With this result and the symmetry of the Fourier transform (Eq. (4.8)) we also find

$$\begin{aligned} {\mathcal {F}}\left\{ \textrm{sign}\right\} (\omega ) = 2\pi \, \textrm{pv} \, \frac{\jmath }{\pi \, (-\omega )} = \textrm{pv} \, \frac{2}{\jmath \omega }\,. \end{aligned}$$

Example 4.9: Fourier transform of $\boldsymbol{\textsf{1}_{+}}$

The Heaviside step function $\textsf{1}_{+}$ can be written as

$$\begin{aligned} \textsf{1}_{+}(t) = \frac{1}{2} \left[ 1 + \textrm{sign}(t) \right] \,. \end{aligned}$$

Its Fourier transform is therefore

$$\begin{aligned} {\mathcal {F}}\left\{ \textsf{1}_{+}(t)\right\} = \frac{1}{2} \left[ 2\pi \, \delta (\omega ) + \textrm{pv} \, \frac{2}{\jmath \omega } \right] = \pi \, \delta + \textrm{pv} \, \frac{1}{\jmath \omega }\,. \end{aligned}$$

From the symmetry of the Fourier transform we also obtain

$$\begin{aligned} {\mathcal {F}}\left\{ \pi \, \delta + \textrm{pv} \, \frac{1}{\jmath t}\right\} = 2\pi \, \textsf{1}_{+}(-\omega ) \end{aligned}$$

or

$$\begin{aligned} {\mathcal {F}}\left\{ \frac{1}{2}\delta + \textrm{pv} \, \frac{\jmath }{2\pi \, t}\right\} = \textsf{1}_{+}(\omega )\,. \end{aligned}$$

5 Periodic Distributions

In this section we investigate the Fourier transform of periodic distributions. Consider first a regular distribution arising from a locally integrable periodic function f. If we introduce a function $f_\sqcap$

$$\begin{aligned} f_\sqcap (t) = \left\{ \begin{array}{ll} f(t) &{} a \le t < a + {\mathcal {T}}\\ 0 &{} \text {otherwise} \end{array} \right. \end{aligned}$$

with a a constant, then f can be expressed as a convolution product

$$\begin{aligned} f(t) = f_\sqcap (t) *\delta _{\mathcal {T}}\,. \end{aligned}$$

(4.15)

The Fourier transform of f can therefore be written as the product of the transforms of $f_\sqcap$ and $\delta _{\mathcal {T}}$, which is well-defined since $f_\sqcap$ has compact support

$$\begin{aligned} {\mathcal {F}}\left\{ f\right\} = \left\langle f_\sqcap ,\text {e}^{-\jmath \omega t} \right\rangle \omega _c \delta _{\omega _c} = \frac{2\pi }{{\mathcal {T}}} \sum _{m=-\infty }^\infty \left\langle f_\sqcap ,\text {e}^{-\jmath m\omega _c t} \right\rangle \delta (\omega - m\omega _c)\,. \end{aligned}$$

From this we see that the Fourier transform of f consists of a train of equally spaced Dirac pulses, each weighted by a numerical coefficient, and that this set of weighting numbers fully characterize it.

If we now represent f as the inverse Fourier transform of ${\mathcal {F}}\left\{ f\right\}$ and make use of the results of Example 4.6, we obtain a trigonometric series

$$\begin{aligned} f(t) {} & {} = \frac{2\pi }{{\mathcal {T}}} \sum _{m=-\infty }^\infty \left\langle f_\sqcap ,\text {e}^{-\jmath m\omega _c t} \right\rangle {\mathcal {F}}^{-1}\{\delta (\omega - m\omega _c)\} \nonumber \\ {} & {} = \frac{1}{{\mathcal {T}}} \sum _{m=-\infty }^\infty \left\langle f_\sqcap ,\text {e}^{-\jmath m\omega _c t} \right\rangle \text {e}^{\jmath m\omega _c t}\,. \end{aligned}$$

(4.16)

called the Fourier series of f. The coefficients are values obtained by evaluating $f_\sqcap$ on indefinitely differentiable periodic functions which are members of ${\mathcal {D}}({\mathbb {T}})$ and the values are identical to the ones obtained by evaluating the distribution $f^\circ \in {\mathcal {D'}}({\mathbb {T}})$ corresponding to f (see Sect. 3.4) on the same functions. Consequently, the above trigonometric series is both a representation of a periodic distribution in ${\mathcal {D'}}({\mathbb {R}})$ as well as that of a distribution in ${\mathcal {D'}}({\mathbb {T}})$.

These arguments can be extended to general periodic distributions without any difficulty so that we have the following general definition of the Fourier series  of a periodic distribution.

Definition 4.5

(Fourier Series) The Fourier series of a distribution $T^\circ \in {\mathcal {D'}}({\mathbb {T}})$, or a periodic distribution $T \in {\mathcal {D'}}({\mathbb {R}})$, is the trigonometric series

$$\begin{aligned} \sum _{m=-\infty }^\infty c_m \text {e}^{\jmath m\omega _c p} \end{aligned}$$

(4.17)

with coefficients

$$\begin{aligned} c_m = \frac{1}{{\mathcal {T}}} \left\langle T^\circ ,\text {e}^{-\jmath m\omega _c p} \right\rangle \,. \end{aligned}$$

(4.18)

The coefficients are called the Fourier coefficients of the series.

The Fourier series is the only trigonometric series that converges to the distribution $T^\circ$ in ${\mathcal {D'}}({\mathbb {T}})$. In fact, if for any $\Phi \in {\mathcal {D}}({\mathbb {T}})$ the series

$$\begin{aligned} \sum _{m=-\infty }^\infty d_m \left\langle \text {e}^{\jmath m\omega _c p},\Phi \right\rangle \end{aligned}$$

does converge, then by putting $\Phi = \text {e}^{-\jmath m\omega _c p}$ and using the orthogonality of trigonometric functions we find that

$$\begin{aligned} \left\langle T^\circ ,\text {e}^{-\jmath m\omega _c p} \right\rangle = {\mathcal {T}}d_m \end{aligned}$$

which shows that the coefficients $d_m$ correspond to the Fourier coefficients of $T^\circ$.

As every distribution, the Fourier series of a distribution can be differentiated term by term. Therefore, if we designate by $c_m(T^\circ )$ the mth Fourier coefficient of the distribution $T^\circ$, we have that

$$\begin{aligned} c_m(D^k T^\circ ) = (\jmath m\omega _c)^k c_m(T^\circ )\,. \end{aligned}$$

(4.19)

A natural question to ask is: How do we know if a certain trigonometric series converges to a periodic distribution? To answer this question first note that the series of numbers

$$\begin{aligned} \sum _{m=1}^\infty \frac{1}{m^2} \end{aligned}$$

is absolutely convergent. Therefore, if the magnitude of the coefficients $|c_m|$, as $m \rightarrow \infty$, are bounded above by $C/|m|^2$, with C a constant, then the series converges to a continuous function f and hence to a distribution. But distributions are always differentiable term by term an arbitrary number of times. Using (4.19) we therefore conclude that, if the magnitude of the coefficients of the series, as $m \rightarrow \infty$ are bound by $C |m|^k$ for some number $k \ge 0$ and a constant C, then the series converges to a distribution.

We derived the Fourier series starting from the Fourier transform and its property that converts convolution into a product. We therefore expect a similar property for the Fourier series. Consider the convolution of two distributions $S^\circ$ and $T^\circ$ with the same period ${\mathcal {T}}$. The Fourier coefficients of the resulting series are

$$\begin{aligned} c_m(S^\circ *T^\circ ) {} & {} = \frac{1}{{\mathcal {T}}} \left\langle S^\circ *T^\circ ,\text {e}^{-\jmath m \omega _c t} \right\rangle \nonumber \\ {} & {} = \frac{1}{{\mathcal {T}}} \left\langle S^\circ (t) \otimes T^\circ (\lambda ),\text {e}^{-\jmath m \omega _c (t + \lambda )} \right\rangle \nonumber \\ {} & {} = \frac{1}{{\mathcal {T}}} \left\langle S^\circ (t),\text {e}^{-\jmath m \omega _c t} \right\rangle \left\langle T^\circ (t),\text {e}^{-\jmath m \omega _c \lambda } \right\rangle \nonumber \\ {} & {} = {\mathcal {T}}\, c_m(S^\circ ) \, c_m(T^\circ )\,. \end{aligned}$$

(4.20)

Consequently the Fourier series of the convolution of $S^\circ$ and $T^\circ$ is

$$\begin{aligned} S^\circ *T^\circ = {\mathcal {T}}\sum _{m=-\infty }^\infty c_m(S^\circ ) \, c_m(T^\circ ) \text {e}^{\jmath m \omega _c t} \end{aligned}$$

(4.21)

and, indeed we see that the Fourier series representation of periodic distributions transforms convolutions into products.

Example 4.10: Fourier series of $\boldsymbol{\delta _{\mathcal {T}}}$

The mth Fourier coefficient of the Dirac comb $\delta _{\mathcal {T}}$ is

$$\begin{aligned} c_m(\delta _{\mathcal {T}}) = \frac{1}{{\mathcal {T}}}\left\langle \delta ^\circ ,\text {e}^{-\jmath m \omega _c t} \right\rangle = \frac{1}{{\mathcal {T}}} \end{aligned}$$

with $\omega _c = 2\pi /{\mathcal {T}}$. Hence, its Fourier series is

$$\begin{aligned} \delta _{\mathcal {T}}= \sum _{m=-\infty }^\infty \frac{1}{{\mathcal {T}}} \, \text {e}^{\jmath m \omega _c t}\,. \end{aligned}$$

If we now compute the convolution of $\delta _{\mathcal {T}}$ with another ${\mathcal {T}}$-periodic distribution T with Fourier coefficients $c_m(T)$, from (4.21) we see that, as expected, $\delta _{\mathcal {T}}$ act as a unit

$$\begin{aligned} c_m(T *\delta _{\mathcal {T}}) = {\mathcal {T}}c_m(\delta _{\mathcal {T}}) \, c_m(T) = c_m(T)\,. \end{aligned}$$

A periodic distribution can be represented as a convolution product between the Dirac comb $\delta _{\mathcal {T}}$ and a distribution different from the one of (4.15). For example, with $\xi _{\mathcal {T}}$ any unitary function we have

$$\begin{aligned} T {} & {} = T \sum _{m=-\infty }^\infty \xi _{\mathcal {T}}(t - m {\mathcal {T}}) \nonumber \\ {} & {} = \sum _{m=-\infty }^\infty T(t - m {\mathcal {T}}) \, \xi _{\mathcal {T}}(t - m {\mathcal {T}}) \nonumber \\ {} & {} =: \sum _{m=-\infty }^\infty S(t - m {\mathcal {T}}) = S *\delta _{\mathcal {T}}\end{aligned}$$

(4.22)

which defines a distribution S whose support is finite and larger than a single period of T. Using this representation we can express the Fourier coefficients and the Fourier transform of T in terms of the one of S as

$$\begin{aligned} \hat{T} {} & {} = \omega _c \, \hat{S} \, \delta _{\omega _c} = \frac{2\pi }{{\mathcal {T}}} \sum _{m=-\infty }^\infty \hat{S}(\omega ) \, \delta (\omega - m \omega _c) \end{aligned}$$

(4.23)

$$\begin{aligned} c_m(T) {} & {} = \frac{\hat{S}(m\omega _c)}{{\mathcal {T}}}\,. \end{aligned}$$

(4.24)

For this reason, if in some calculation we obtain the Fourier transform of a signal in this form, with $\hat{S}$ the transform of a known non-periodic distribution, then we can immediately write T in terms of S as in (4.22).

We close this section with a property that is the counterpart of (4.10) for the Fourier coefficients of a real periodic distribution

$$\begin{aligned} \overline{c}_m = c_{-m}\,. \end{aligned}$$

(4.25)

6 Extension to Several Variables

The Fourier transform can be extended to functions of several variables by transforming each variable individually. That is, if f is an integrable function on ${\mathbb {R}}^n$, then we can apply the one-dimensional Fourier transform to each variable individually, keeping the other ones constant. After performing this operation with respect to each variable in turns, we obtain the following expression which defines of the n-dimensional Fourier transform

To shorten the notation we will write

$$\begin{aligned} \hat{f}(\omega ) = {\mathcal {F}}\left\{ f\right\} (\omega ) = \int _{{\mathbb {R}}^n} f(\tau ) \, \text {e}^{-\jmath \left( \omega ,\tau \right) } \, d^n\tau \end{aligned}$$

(4.26)

with $\tau , \omega \in {\mathbb {R}}^n$ and

$$\begin{aligned} \left( \omega ,\tau \right) :=\sum _{m=1}^n \omega _m \tau _m\,. \end{aligned}$$

The n-dimensional inverse Fourier transform can be derived with the same procedure, and we obtain the following definition

$$\begin{aligned} {\mathcal {F}}^{-1}\{f\}(\tau ) :=\frac{1}{(2\pi )^n} \int _{{\mathbb {R}}^n} f(\omega ) \, \text {e}^{\jmath \left( \omega ,\tau \right) } \, d^n\omega \,. \end{aligned}$$

(4.27)

With these definitions it’s easy to see that our definition of Fourier transform for tempered distributions remains valid for $n>1$ as well. All properties carry over in similar form. For example, looking back at the derivation of the symmetry relation given by (4.8), we see that in the n-dimensional case it becomes

(4.28)

The only difference from the one dimensional case is the fact that the factor of $2\pi$ becomes $(2\pi )^n$. This happens to all properties involving factors of $2\pi$.

The most important convolution property (4.13) remains unchanged, as can easily be verified by inspecting the derivation for the one-dimensional case.

Before proceeding, it’s convenient to extend the multi-index notation that up to now we only used in conjunction with the differential operator. Let a be an n-tuple in ${\mathbb {C}}^n$ and k a multi-index that we allow to include negative numbers $(k_1,\ldots ,k_n) \in {\mathbb {Z}}^n$. Then we can define

$$\begin{aligned} \begin{array}{rcll} a^k &{}:=&{} a_1^{k_1} \ldots a_n^{k_n} &{} \text {(exponentiation)} \\ k \, a &{}:=&{} (k_1 a_1,\ldots ,k_n a_n) &{} \text {(direct product)}\\ \sum _{k = l_l}^{l_u} f_k &{}:=&{} \sum \limits _{k_1 = l_1}^{u_1} \ldots \sum \limits _{k_n = l_n}^{u_n} f_{k_1,\ldots ,k_n} &{} \text {(summation)} \end{array} \end{aligned}$$

with $f_k$ some function parameterized by the multi-index k and l, u lower resp. upper multi-indices. If in a summation we write integer numbers instead of l and u, we intend multi-indices equal to that number in every position.

We can introduce a multi-index notation for the factorial as well. However, this only makes sense for tuples of natural numbers $k \in {\mathbb {N}}^n$

$$\begin{aligned} k! :=\prod _{i=1}^n (k_i)! . \end{aligned}$$

Example 4.11: Fourier transform of $\boldsymbol{\delta }$

In Example 3.1 we saw that the $\delta$ distribution in ${\mathcal {D'}}({\mathbb {R}}^n)$ is the tensor product of one dimensional $\delta$’s. Hence, with $\tau ,\lambda \in {\mathbb {R}}^n$ and the results of Example 4.5 the Fourier transform of the n-dimensional shifted $\delta$ becomes

$$\begin{aligned} {\mathcal {F}}\left\{ \delta (\tau - \lambda )\right\} = \text {e}^{-\jmath \left( \omega ,\tau \right) } \end{aligned}$$

and it’s partial derivative

$$\begin{aligned} {\mathcal {F}}\left\{ D_i\delta (\tau )\right\} = (\jmath \omega _i) \qquad i = 1,\ldots ,n\,. \end{aligned}$$

Using the multi-index notation, the higher order partial derivatives can be conveniently expressed as

$$\begin{aligned} {\mathcal {F}}\left\{ D^k\delta (\tau )\right\} = (\jmath \omega )^k\,. \end{aligned}$$

Using the n-dimensional symmetry relation given by (4.28) we also immediately find

$$\begin{aligned} {\mathcal {F}}\left\{ \text {e}^{\jmath \left( \omega _c,\tau \right) }\right\} = (2\pi )^n \, \delta (\omega - \omega _c) \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}\left\{ (-\jmath \tau )^k\right\} = (2\pi )^n D^k\delta (\omega )\,. \end{aligned}$$

As in the one-dimensional case, the other properties of the n-dimensional Fourier transform are immediate consequences of the convolution property and the convolution and transform of the $\delta$ distribution.

A periodic function on ${\mathbb {R}}^n$ is a function that is periodic in each independent variable individually, that is, such that there are positive numbers ${\mathcal {T}}_i$ for $i=1,\ldots ,n$, called the period of the ith independent variable, so that

$$\begin{aligned} f(\tau _1,\ldots ,\tau _i + {\mathcal {T}}_i,\ldots ,\tau _n) = f(\tau _1,\ldots ,\tau _i,\ldots ,\tau _n)\,. \end{aligned}$$

This extension of the concept of a periodic function to higher dimensions permits us to widen the definition of periodic distributions (3.23) on test function of higher dimensions ${\mathcal {D}}({\mathbb {R}}^n)$ in a straightforward way

From this follows without any difficulty an extension of the second Definition 3.5 as well.

The n-dimensional Fourier series of a periodic distribution $T \in {\mathcal {D'}}({\mathbb {R}}^n)$ is

$$\begin{aligned} \sum _{k=-\infty }^\infty c_k(T) \, \text {e}^{\jmath \left( k\omega _c,\tau \right) } \end{aligned}$$

with k an n-dimensional multi-index, $\omega _c$ the n-tuple $(2\pi /{\mathcal {T}}_1,\ldots ,2\pi /{\mathcal {T}}_n)$ and $c_k(T)$ the Fourier coefficients

$$\begin{aligned} c_k(T) = \frac{1}{{\mathcal {T}}_1 \cdot \cdots \cdot {\mathcal {T}}_n}\left\langle T^\circ ,\text {e}^{-\jmath \left( k\omega _c,\tau \right) } \right\rangle \,. \end{aligned}$$