Linear Differential Equations and Related Continuous LTI Systems

Abstract

In this paper, we consider the problem, usual in analog signal processing, to find a continuous linear time-invariant system related to a linear differential equation $P(D)x = Q(D)f$, i.e. a system $\mathscr {L}$ such that for every input signal f yields an output $\mathscr {L}(f)$ which verifies $P(D)\mathscr {L}(f ) = Q(D)f$. We give a systematic theoretical analysis of the existence and uniqueness of such systems (both causal and non-causal ones) defined on ${L^{p}}$ functions and ${\mathscr {D'}_{L^{p}}}$ distributions (input spaces which include signals with not necessarily left-bounded support), for every p. More precisely, by finding all their possible impulse responses, we characterise all these systems apart two pathologies arising when $p = \infty$. Finally, we give necessary and sufficient conditions on P, Q for causality and stability of the systems. As an application, we consider the problem of finding the inverse of a causal continuous linear time-invariant system, defined on ${L^{p}}$, related to a simple differential equation. We also show a digital simulation of this inverse system.

Appendix: Uniformly Almost Periodic Functions

Let $f(t):\mathbb {R}\rightarrow \mathbb {C}$ be a function, and let $\varepsilon \not =0$ be a real number. In this Section, we call almost period of f(t) relative to $\varepsilon$ a real number $T > 0$ such that for every $t_{0}\in \mathbb {R}$, there exists $\tau \in \left[ t_{0},t_{0} + T\right]$ satisfying

$$\begin{aligned} \left| \,f(t + \tau ) - f(t)\,\right| < \varepsilon \quad \text {for every } t\in \mathbb {R}\end{aligned}$$

The function f(t) is called uniformly almost periodic if $f(t)\in {C^{0}}$ and for every $\varepsilon > 0$ there exists an almost period T of f(t) relative to $\varepsilon$.

Immediate consequences of the definition are

• (a) Every continuous periodic function is uniformly almost periodic

• (b) If f(t) is uniformly almost periodic then for every $\lambda \in \mathbb {R}$, $f(\lambda t)$ and $f(t + \lambda )$ are uniformly almost periodic

Moreover, the following results are well know:

• (c) The set of uniformly almost periodic functions is a vector subspace of ${C^{0}}$ (see [1, Theorem 6 and Theorem 12])

• (d) Every linear combination of continuous periodic functions is uniformly almost periodic (follows by items (a) and (c))

• (e) A uniformly almost periodic function is bounded and uniformly continuous (see [1, Theorems 4 and 5])

• (f) If f(t) is uniformly almost periodic then $\mathrm{Re}f(t)$ and $\mathrm{Im}f(t)$ are uniformly almost periodic (follows by [1, Theorem 6] and item (c))

The following theorem proves that every non null uniformly almost periodic function cannot be null at infinity.

Theorem 7

Let f(t) be a uniformly almost periodic function; the following statements are equivalent:

• a) ${\displaystyle \lim _{t \rightarrow +\infty } }{f(t)} = 0$

• b) ${\displaystyle \lim _{t \rightarrow -\infty } }{f(t)} = 0$

• c) $f(t) = 0$ for every $t\in \mathbb {R}$

Proof

a)$\Rightarrow$c). Let $n > 1$. By the assumption there exists a real number $t_{n} > 0$ such that: $|\,f(t_{n} + t)\,| < 1/n$ for every $t > 0$.

Let $T_{n}$ be an almost period of f(t) relative to 1 / n. Then, there exists $\tau _{n}\in [-n-t_{n}-T_{n},-n-t_{n} ]$ such that: $|\,f(t + \tau _{n}) - f(t)\,| < 1/n$ for every $t\in \mathbb {R}$. Writing $\tau _{n} = -n-t_{n}-\sigma _{n}$, with $0\leqslant \sigma _{n}\leqslant T_{n}$, we obtain $|\,f(t-n-t_{n}-\sigma _{n}) - f(t)\,| < 1/n$ for every $t\in \mathbb {R}$. Substituting t with $t_{n} + t$ we obtain: $|\,f(-n-\sigma _{n}+t) - f(t_{n} + t)\,| < 1/n$ for every $t\in \mathbb {R}$, and hence $|\,f(-n-\sigma _{n}+t)\,| < |\,f(t_{n}+t)\,| + 1/n$ for every $t\in \mathbb {R}$.

As a consequence it is $|\,f(-n-\sigma _{n}+t)\,| < 2/n$ for every $t > 0$, and hence $|\,f(t)\,| < 2/n$ for every $t > -n$.

b)$\Rightarrow$c). Let $g(t) = f(-t)$ and apply the previous result to g(t).

Obviously c)$\Rightarrow$a), c)$\Rightarrow$b). $\square$

The following theorem proves that every regularization of a uniformly almost periodic function is uniformly almost periodic.

Theorem 8

Let f(t) be a uniformly almost periodic function, and let $\varphi (t)\in {\mathscr {D}}$. Then $(f*\varphi )(t)$ is uniformly almost periodic.

Proof

If $\varphi (t) = 0$, the statement is obvious. Let $\varphi (t) \not = 0$, and let $\mu = \int {}{}{|\varphi |} > 0$.

Let $\varepsilon > 0$, let $T > 0$ be an almost period of f(t) relative to $\varepsilon /(2\mu )$, and let $t_{0}\in \mathbb {R}$.

There exists $\tau \in [t_{0}, t_{0} + T]$ such that: $|\,f(t + \tau ) - f(t)\,| < \varepsilon /(2\mu )$ for every $t\in \mathbb {R}$. Write $f_{\tau }(t) = f(t + \tau )$. For every $t\in \mathbb {R}$ it is

$$\begin{aligned}&{|\,(f *\varphi )(t + \tau ) - (f *\varphi )(t)\,| = |\,(f_{\tau } *\varphi )(t) - (f *\varphi )(t)\,|} \nonumber \\&\quad = |\,((f_{\tau } - f) *\varphi )(t)\,| = \left| {\displaystyle \int _{}^{}(f_{\tau } - f)(x)\, \varphi (t - x)\, \mathrm{d}x}\,\right| \nonumber \\&\quad \leqslant {\displaystyle \int _{}^{}|\,f(x + \tau ) - f(x)\,|\,|\,\varphi (t - x)\,|\,\mathrm{d}x}\leqslant (\varepsilon /(2\mu )){\displaystyle \int _{}^{}|\,\varphi \,|} = \varepsilon /2 < \varepsilon \end{aligned}$$

As a consequence T is an almost period of $(f *\varphi )(t)$ relative to $\varepsilon$. $\square$

As corollaries of Theorems 7 and 8, we can now state the results used in the proofs of the theorems of Sects. 2 and 3.

Corollary 1

Let $\omega _{1},\dots , \omega _{\nu }\in \mathbb {R}$ be pairwise distinct, and $c_{1},\dots , c_{\nu }\in \mathbb {C}$ be not all equal to 0. Then, there exists $\mu > 0$ and two sequences $\tau _{k}', \tau _{k}''\in \mathbb {R}$ such that

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}'} = +\infty ,\quad \left| {\displaystyle \sum _{l = 1}^{\nu }c_{l}e^{i\omega _{l}\tau _{k}'}}\,\right| > \mu \quad \text {for every } k \end{aligned}$$

and

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}''} = - \infty ,\quad \left| {\displaystyle \sum _{l = 1}^{\nu }c_{l}e^{i\omega _{l}\tau _{k}''}}\,\right| > \mu \quad \text {for every } k \end{aligned}$$

Proof

Simply observe that $\sum _{l = 1}^{\nu }c_{l}e^{i\omega _{l}t}$ is uniformly almost periodic (as a linear combination of continuous periodic functions), and nonzero (since the family $e^{i\omega _{l}t}, l = 1,\dots ,\nu$ is linearly independent); then apply Theorem 7. $\square$

Corollary 2

Let $\omega _{1},\dots ,\omega _{\nu }\in \mathbb {R}$ be pairwise distinct, and let $c_{1},\dots ,c_{\nu }\in \mathbb {C}$. The following statements are equivalent

• a) ${\displaystyle \sum _{l = 1}^{\nu }c_{l}e^{i\omega _{l}t}}$ is a distribution null at $\infty$

• b) ${\displaystyle \sum _{l = 1}^{\nu }c_{l}e^{i\omega _{l}t}}$ is a distribution null at $-\infty$

• c) $c_{1} = \dots = c_{\nu } = 0$

Proof

Obviously a)$\Rightarrow$b), and c)$\Rightarrow$a).

b)$\Rightarrow$c). Let $B\in \mathbb {R}$ be such that $0< B < \min \{\frac{\pi }{4\,|\,\omega _{l}\,|}: l \text {such that } \omega _{l} \not = 0\}$ and let $\varphi (t)\in {\mathscr {D}}$ be such that ${\mathrm{supp~}}\varphi \subset (-B,B)$, $\varphi (t) \geqslant 0$ for every $t\in \mathbb {R}$, $\int _{}^{}\varphi \not = 0$. A straightforward argument proves that ${\hat{\varphi }}(\omega _{l}) \not = 0$ for every l (here ${\hat{\varphi }}(\omega )$ denotes the Fourier Transform of $\varphi (t)$).

Observe that $(\sum _{l = 1}^{\nu } c_{l}e^{i\omega _{l}t}) *\varphi (t) = \sum _{l = 1}^{\nu } c_{l}\,{\hat{\varphi }}(\omega _{l})\,e^{i\omega _{l}t}$ is a function null at $-\infty$ in the usual sense. As a consequence, by Corollary 1, for every l we have $c_{l}\,{\hat{\varphi }}(\omega _{l}) = 0$, and hence $c_{l} = 0$. $\square$

Corollary 3

Let $\omega _{1},\dots ,\omega _{\nu }\in \mathbb {R}$ be pairwise distinct, and $M_{1}(t),\dots ,M_{\nu }(t)$ be polynomial functions not all constant. Then, there exist two sequences $\tau _{k}', \tau _{k}''\in \mathbb {R}$ such that

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}'} = +\infty ,\quad {\displaystyle \lim _{k \rightarrow \infty } }{\left| {\displaystyle \sum _{l = 1}^{\nu }M_{l}(\tau _{k}')\,e^{i\omega _{l}\tau _{k}'}}\right| } = +\infty \end{aligned}$$

and

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}''} = -\infty ,\quad {\displaystyle \lim _{k \rightarrow \infty } }{\left| {\displaystyle \sum _{l = 1}^{\nu }M_{l}(\tau _{k}'')\,e^{i\omega _{l}\tau _{k}''}}\right| } = +\infty \end{aligned}$$

Proof

Write the polynomial functions $M_{l}(t)$ in the form $M_{l}(t) = \sum _{h = 0}^{p} m_{lh}t^{h}$ with $p \geqslant 1$ and at least one of the coefficients $m_{1p},\dots ,m_{\nu p}$ non null. By Corollary 1, there exists $\mu > 0$ and two sequences $\tau _{k}', \tau _{k}''\in \mathbb {R}$ such that

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}'} = +\infty ,\quad \left| {\displaystyle \sum _{l = 1}^{\nu }m_{lp}e^{i\omega _{l}\tau _{k}'}}\,\right| > \mu \quad \text {for every } k \end{aligned}$$

and

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}''} = -\infty ,\quad \left| {\displaystyle \sum _{l = 1}^{\nu }m_{lp}e^{i\omega _{l}\tau _{k}''}}\,\right| > \mu \quad \text {for every } k \end{aligned}$$

Then observe that for every k it is

$$\begin{aligned} {\displaystyle \sum _{l = 1}^{\nu }M_{l}(\tau _{k}^{\prime })\,e^{i\omega _{l}\tau _{k}^{\prime }}} = (\tau _{k}^{\prime })^{p}\left( \alpha _{k}^{\prime } + {\displaystyle \sum _{l = 1}^{\nu }m_{lp}e^{i\omega _{l}\tau _{k}^{\prime }}}\right) \end{aligned}$$

and

$$\begin{aligned} {\displaystyle \sum _{l = 1}^{\nu }M_{l}(\tau _{k}^{\prime \prime })\, e^{i\omega _{l}\tau _{k}^{\prime \prime }}} = (\tau _{k}^{\prime \prime })^{p}\left( \alpha _{k}^{\prime \prime } + {\displaystyle \sum _{l = 1}^{\nu }m_{lp}e^{i\omega _{l}\tau _{k}^{\prime \prime }}}\right) \end{aligned}$$

where ${\displaystyle \lim _{k \rightarrow \infty } }{\alpha _{k}^{\prime }} = 0$ and ${\displaystyle \lim _{k \rightarrow \infty } }{\alpha _{k}^{\prime \prime }} = 0$. $\square$

Corollary 4

Let $s_{1} = {\sigma _{1}+i\omega _{1}},\dots ,s_{\nu } = {\sigma _{\nu }+i\omega _{\nu }}$ be pairwise distinct complex numbers, $M_{1}(t),\dots ,$$M_{\nu }(t)$ be polynomial functions not all null, and $P_{0}(t)$ be a polynomial function. If $\sigma _{l} \not = 0$ for every $l= 1,..., \nu$ then

• a) ${\displaystyle \sum _{l = 1}^{\nu }M_{l}(t)e^{s_{l}t}} + P_{0}(t)$ is not a slowly increasing function in the usual sense

• b) ${\displaystyle \sum _{l = 1}^{\nu }M_{l}(t)e^{s_{l}t}}\not \in {\mathscr {S'}}$

Proof

Changing, if necessary, the family of indices, we may assume $M_{l}(t)$ not null for every l.

a) Assume first that there exists l such that $\sigma _{l} > 0$, and let $\sigma = \max \{\,\sigma _{1},\dots ,\sigma _{\nu }\,\}$$> 0.$ Then it is

$$\begin{aligned} {\displaystyle \sum _{l = 1}^{\nu }M_{l}(t)e^{s_{l}t}} + P_{0}(t) = e^{\sigma t}\left( \alpha (t) + {\displaystyle \sum _{l:\,\sigma _{l} = \sigma }^{}M_{l}(t)e^{i\omega _{l}t}}\right) \end{aligned}$$

where $\lim _{t \rightarrow +\infty }\alpha (t) = 0$. By Corollaries 1 and 3, there exist $\mu > 0$ and a sequence $\tau _{k}\in \mathbb {R}$ such that

$$\begin{aligned} {\displaystyle \lim _{k \rightarrow \infty } }{\tau _{k}} = +\infty ,\quad \left| {\displaystyle \sum _{l:\,\sigma _{l} = \sigma }^{}M_{l}(\tau _{k})e^{i\omega _{l}\tau _{k}}}\,\right| > \mu \quad \text {for every } k \end{aligned}$$

As a consequence $|\sum _{l = 1}^{\nu } M_{l}(t)e^{s_{l}t} + P_{0}(t)|$ cannot be bounded by any $|\,F(t)\,|$ where F(t) is a polynomial function.

Assume now that $\sigma _{l} < 0$ for every l, and let $\varPhi (t) = \sum _{l = 1}^{\nu } M_{l}(-t)e^{-s_{l}t} + P_{0}(-t)$. By the previous result $\varPhi (t)$ is not slowly increasing; as a consequence, also $\varPhi (-t) = \sum _{l = 1}^{\nu } M_{l}(t)e^{s_{l}t} + P_{0}(t)$ it is not.

b) Let $f(t) = \sum _{l = 1}^{\nu } M_{l}(t)e^{s_{l}t}$. By [17, Chapter VII, Theorem VI] $f(t)\in {\mathscr {S'}}$ if and only if there exist $n\in \mathbb {N}, g(t)\in {C^{0}}$ such that g(t) is slowly increasing and $f(t) = g^{(n)}(t)$. It is well known that every $g(t)\in {\mathscr {D'}}$ such that $f(t) = g^{(n)}(t)$ may be written in the form $g(t) = \sum _{l = 1}^{\nu } N_{l}(t)e^{s_{l}t} + Q_{0}(t)$ where $N_{l}(t)$ and $Q_{0}(t)$ are polynomial functions. Obviously there exists $l \in \{\,1,\ldots ,\nu \,\}$ such that $N_{l}(t) \not = 0$. By a), g(t) cannot be slowly increasing. $\square$

Corollary 5

Let

$$\begin{aligned} F(t) = {\displaystyle \sum _{l = 0}^{p}t^{l}f_{l}(t)} \end{aligned}$$

where all the $f_{l}(t)$ are uniformly almost periodic functions, $p \geqslant 1, f_{p}(t) \not = 0$. Then $F(t), F(t)H(t), F(t)H(-t)\not \in {\mathscr {D'}_{L^{\infty }}}$.

Proof

Changing if necessary F(t) with iF(t), we may assume $\mathrm{Re}\,f_{p}(t) \not = 0$. Changing if necessary F(t) with $-F(t)$, by Theorem 7 we may assume that there exist a real number $\mu > 0$ and a sequence $\tau _{k}\in \mathbb {R}$ such that: $\lim _{k\rightarrow \infty }{\tau _{k}} = +\infty$ and $\mathrm{Re}\,f_{p}(\tau _{k}) > \mu$ for every k.

Let T be an almost period of $\mathrm{Re}\,f_{p}(t)$ relative to $\frac{1}{8}\mu$.

For every $\tau _{k}$, there exists a real number $-\tau _{k} + \sigma _{k}\in [-\tau _{k}, -\tau _{k} + T]$ (i.e. a real number $\sigma _{k}\in [0,T]$) such that: $|\,\mathrm{Re}\,f_{p}(t - \tau _{k} + \sigma _{k}) - \mathrm{Re}\,f_{p}(t)\,| < \textstyle \frac{1}{8}\mu$ for every $t\in \mathbb {R}$ and hence, changing t with $t + \tau _{k}$, such that

$$\begin{aligned} |\,\mathrm{Re}\,f_{p}(t + \sigma _{k}) - \mathrm{Re}\,f_{p}(t + \tau _{k})\,| < \textstyle \frac{1}{8}\mu \qquad \text {for every } t\in \mathbb {R}\end{aligned}$$

(A.1)

Since every $\sigma _{k}\in [0,T]$, changing if necessary the sequence $\tau _{k}$ with a subsequence, we may assume that there exists $\sigma \in [0,T]$ such that $\lim _{k\rightarrow \infty }\sigma _{k} = \sigma$.

Since $\mathrm{Re}\,f_{p}(t)$ is uniformly continuous, there exists a real number $\zeta > 0$ such that for every pair $( t', t'')$ such that $|\,t'' - t'\,| < \zeta$ it is $|\,\mathrm{Re}\,f_{p}(t'') - \mathrm{Re}\,f_{p}(t')\,| < \frac{1}{8}\mu$. Changing if necessary the sequence $\tau _{k}$ with a subsequence, we may assume that $|\,\sigma _{k} - \sigma \,| < \zeta$ for every $\tau _{k}$, and hence

$$\begin{aligned} |\,\mathrm{Re}\,f_{p}(t + \sigma _{k}) - \mathrm{Re}\,f_{p}(t + \sigma )\,| < \textstyle \frac{1}{8}\mu \qquad \text {for every } t\in \mathbb {R}\end{aligned}$$

(A.2)

As a consequence of Eqs. (A.1) and (A.2) it is

$$\begin{aligned}&|\,\mathrm{Re}\,f_{p}(t + \sigma ) - \mathrm{Re}\,f_{p}(t + \tau _{k})\,| = |\,\mathrm{Re}\,f_{p}(t + \sigma ) - \mathrm{Re}\,f_{p}(t + \sigma _{k}) \nonumber \\&\quad + \mathrm{Re}\,f_{p}(t + \sigma _{k}) - \mathrm{Re}\,f_{p}(t + \tau _{k})\,| < 2\,\textstyle \frac{1}{8}\mu = \frac{1}{4}\mu \qquad \text {for every } t\in \mathbb {R}\end{aligned}$$

(A.3)

Taking $t = 0$ in Eq. (A.3) we obtain $|\mathrm{Re}\,f_{p}(\sigma ) - \mathrm{Re}\,f_{p}(\tau _{k})| < \frac{1}{4}\mu$. Since we have $\mathrm{Re}\,f_{p}(\tau _{k}) > \mu$, it is $\mathrm{Re}\,f_{p}(\sigma ) > \mu - \frac{1}{4}\mu = \frac{3}{4}\mu$. Then there exists a real number $B > 0$ such that: $\mathrm{Re}\,f_{p}(t + \sigma ) > \textstyle \frac{3}{4}\mu$ for every $t\in (-B,B)$. As a consequence of Eq. (A.3), it is also

$$\begin{aligned} \mathrm{Re}\,f_{p}(t + \tau _{k}) > \textstyle \frac{3}{4}\mu - \frac{1}{4}\mu = \frac{1}{2}\mu \quad \text {for every } t\in (-B,B) \end{aligned}$$

(A.4)

Last, observe that, for every $t \not = 0$, F(t) may be uniquely written in the form $F(t) = t^{p}(\,\psi (t) + f_{p}(t)\,)$ with $\psi (t)$ continuous on $\mathbb {R}\setminus \{\,0\,\}$, and such that there exists a real number $C > 0$ such that: $|\,\psi (t)\,| < \textstyle \frac{1}{4}\mu$ for every $|\,t\,| > C$.

Let $\varPhi (t) = F(t)H(t).$

Let $\varphi (t)\in {\mathscr {D}}$ be such that ${\mathrm{supp~}}\varphi \subset (-B,B)$, $\varphi (t) \geqslant 0$ for every $t\in \mathbb {R}$, $\int _{}^{}\varphi = 1$. For every $\tau _{k} > C + B$ it is

$$\begin{aligned} \mathrm{Re}\,(\varPhi *\varphi )(\tau _{k})= & {} {\displaystyle \int _{\tau _{k} - B}^{\tau _{k} + B}t^{p}\big (\mathrm{Re}\,\psi (t) + \mathrm{Re}\,f_{p}(t)\big ) \varphi (\tau _{k} - t)\,\mathrm{d}t} \\= & {} {\displaystyle \int _{-B}^{B}(\tau _{k} + t)^{p}\big (\mathrm{Re}\,\psi (\tau _{k} + t) + \mathrm{Re}\,f_{p}(\tau _{k} + t)\big ) \varphi (-t)\,\mathrm{d}t} \end{aligned}$$

Since for every $t\in (-B,B)$ it is: $\varphi (-t) \geqslant 0$, $(\tau _{k} + t)^{p} \geqslant (\tau _{k} - B)^{p} > 0$ and (since $\tau _{k} + t > C$) $\mathrm{Re}\,\psi (\tau _{k} + t) + \mathrm{Re}\,f_{p}(\tau _{k} + t)> -\frac{1}{4}\mu + \frac{1}{2}\mu = \frac{1}{4}\mu > 0$, then

$$\begin{aligned} \mathrm{Re}\,(\varPhi *\varphi )(\tau _{k}) \geqslant (\tau _{k} - B)^{p} \,\textstyle \frac{1}{4} \mu {\displaystyle \int _{}^{}\varphi } = \frac{1}{4}\mu \,(\tau _{k} - B)^{p} \end{aligned}$$

Then $(\varPhi *\varphi )(t)$ is not bounded in $\mathbb {R}$, and hence $\varPhi (t) = F(t)H(t)\not \in {\mathscr {D'}_{L^{\infty }}}$.

The proof for F(t) and $F(t)H(-t)$ are straightforward consequences. $\square$

Corollary 6

Let

$$\begin{aligned} F(t) = {\displaystyle \sum _{l = 0}^{p}t^{l}f_{l}(t)} ,\quad G(t) = {\displaystyle \sum _{l = 0}^{q}t^{l}g_{l}(t)} \end{aligned}$$

where all the $f_{l}(t)$ and $g_{l}(t)$ are uniformly almost periodic functions. The following statements are equivalent:

• a) $F(t)H(t) + G(t)\in {\mathscr {D'}_{L^{\infty }}}$

• b) For every $l \not = 0$ it is $f_{l}(t) = 0$ and $g_{l}(t) = 0$

Proof

a)$\Rightarrow$b). Assume for a contradiction that $g_{l}(t) \not = 0$ for some $l \not = 0$. The argument adopted in the Proof of Corollary 5 may be used to prove that $F(t)H(t) + G(t)\not \in {\mathscr {D'}_{L^{\infty }}}$ (simply use a sequence $\tau _{k}$ such that $\lim _{k\rightarrow \infty }\tau _{k} = -\infty$) which is absurd. Hence, it is $g_{l}(t) = 0$ for every $l \not = 0$. In particular, $G(t) = g_{0}(t)$ is a uniformly almost periodic function, and hence $G(t)\in {\mathscr {D'}_{L^{\infty }}}$.

As a consequence $F(t)H(t)\in {\mathscr {D'}_{L^{\infty }}}$, and hence by Corollary 5 it is $f_{l}(t) = 0$ for every $l \not = 0$.

b)$\Rightarrow$a). The function $F(t)H(t) + G(t)$ is locally integrable and bounded; hence $F(t)H(t) + G(t)\in {\mathscr {D'}_{L^{\infty }}}$. $\square$

Corollary 7

Let F(t) and G(t) be uniformly almost periodic functions. If $F(t)H(t) + G(t)\in {\mathscr {\dot{D}'}_{L^{\infty }}}$, then $F(t) = G(t) = 0$.

Proof

Let $\varPhi (t) = F(t)H(t) + G(t)$, $\varphi (t)\in {\mathscr {D}}$, and $\widetilde{\varphi }(t) = \varphi (-t)$. By the argument pointed out in the last paragraph of [17, Chapter VI, section 8] it is $\lim _{|\,t\,|\rightarrow \infty }(\varPhi *\widetilde{\varphi })(t) = 0$. Obviously, there exists $t_{0} < 0$ such that for every $t < t_{0}$ it is $(\varPhi *\widetilde{\varphi })(t) = (G *\widetilde{\varphi })(t)$, hence $\lim _{t\rightarrow -\infty }(G *\widetilde{\varphi })(t) = 0$.

By Theorem 8, $(G *\widetilde{\varphi })(t)$ is uniformly almost periodic, hence by Theorem 7, $(G *\widetilde{\varphi })(t) = 0$ for every $t\in \mathbb {R}$; in particular

$$\begin{aligned} {\left\langle G(t),\varphi (t) \right\rangle } = \int _{}^{} G(\tau )\widetilde{\varphi }(0 - \tau )\,\mathrm{d}\tau = (G *\widetilde{\varphi })(0) = 0 \end{aligned}$$

Hence, for every $\varphi \in {\mathscr {D}}$ it is ${\left\langle G(t),\varphi (t) \right\rangle } = 0$. As a consequence $G(t) = 0$ and $\varPhi (t) = F(t)H(t)$.

Let again $\varphi (t)\in {\mathscr {D}}$, and $\widetilde{\varphi }(t)=\varphi (-t)$. Observe that there exists $t_{0} > 0$ such that for every $t > t_{0}$ it is $(\varPhi *\widetilde{\varphi })(t) = (F *\widetilde{\varphi })(t)$, hence $\lim _{t\rightarrow +\infty }(F *\widetilde{\varphi })(t) = 0$. As a consequence, the argument previously applied to G(t) proves that $F(t) = 0$. $\square$