A Note on BIBO Stability^††thanks: The research leading to these results has received funding from the Swiss National Science Foundation under Grant 200020-162343/1.

The scope of our mathematical statements relies on Schwartz’ famous kernel theorem [5, 16] which delineates the complete class of linear operators that continuously map ${\mathcal{D}}(\mathbb{R})\to{\mathcal{D}}^{\prime}(\mathbb{R})$. We recall that ${\mathcal{D}}(\mathbb{R})={\mathcal{C}}_{\rm c}^{\infty}(\mathbb{R})$ is the space of smooth and compactly supported test functions equipped with the usual Schwartz topology⁴⁴4A sequence of functions $\varphi_{k}\in{\mathcal{D}}(\mathbb{R})$ is said to converge to $\varphi$ in ${\mathcal{D}}(\mathbb{R})$ if (i) there exists a compact domain $\mathbb{F}$ that includes the support of $\varphi$ and of all $\varphi_{k}$, and (ii) $\|\varphi_{k}-\varphi\|_{n}\to 0$ for all $n\in\mathbb{N}$, where $\|\varphi\|_{n}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|{\rm D}^{n}\varphi\|_{L_{\infty}}$ with ${\rm D}^{n}:{\mathcal{D}}(\mathbb{R})\to{\mathcal{D}}(\mathbb{R})$ the $n$th derivative operator.. Its topological dual ${\mathcal{D}}^{\prime}(\mathbb{R})$ is the space of generalized functions also known as distributions. In essence, a distribution $f\in{\mathcal{D}}^{\prime}(\mathbb{R})$ is a linear map—more precisely, a continuous linear functional—that assigns a real number to each test function $\varphi\in{\mathcal{D}}(\mathbb{R})$; this is denoted by $f:\varphi\mapsto\langle f,\varphi\rangle$. For instance, the definition of Dirac’s impulse as a distribution is $\delta:\varphi\mapsto\langle\delta,\varphi\rangle\stackrel{{\scriptstyle\vartriangle}}{{=}}\varphi(0)$.

Beside linearity, the property that defines an LSI operator is ${\mathrm{T}}_{\rm LSI}\{\varphi(\cdot-t_{0})\}(t)={\mathrm{T}}_{\rm LSI}\{\varphi\}(t-t_{0})$ for any $t_{0}\in\mathbb{R}$. Schwartz’ theorem then tells us that there is a one-to-one correspondence between continuous LSI operators ${\mathcal{D}}(\mathbb{R})\to C(\mathbb{R})$ and distributions, with the defining distribution $h\in{\mathcal{D}}^{\prime}(\mathbb{R})$ being the impulse response of the operator. The relevant space of continuous functions here is $C(\mathbb{R})$ with the topology of uniform convergence over compact sets, which involves the system of seminorms $\|f\|_{N}=\sup_{|t|\leq N}|f(t)|,N\in\mathbb{N}$. The latter is an extended functional setup that tolerates arbitrary growth at infinity.

Then, depending on the decay properties of $h$, it is generally possible to extend the domain of the convolution operator ${\mathrm{T}}_{h}$ to some appropriate Banach space according to the procedure described in Section IV. For instance, if $h\in L_{1}(\mathbb{R})$, then ${\mathrm{T}}_{h}$ has a continuous extension $L_{\infty}(\mathbb{R})\to C_{\rm b}(\mathbb{R})\subset C(\mathbb{R})$ that coincides with the classical definition given by (1).

Finally, we note that, for the cases where the Dirac impulse $\delta$ is in the domain of the extended operator (for instance, when $h\in C(\mathbb{R})$), the distributional definition of the convolution given by (6) yields ${\mathrm{T}}_{h}\{\delta\}=h\ast\delta=h$, which explains the term “impulse response.”

In order to investigate the issue of BIBO stability, it is helpful to describe the boundedness and continuity properties of functions via their inclusion in appropriate Banach subspaces of ${\mathcal{D}}^{\prime}(\mathbb{R})$. The three relevant function spaces are

The central space consists of the subset of bounded functions that are continuous:

It is a classical example of Banach space—a complete normed vector space [12]. The smaller space $C_{0}(\mathbb{R})$, which is also equipped with the ${\sup}$-norm, imposes the additional constraint that $f(t)$ should vanish at $t=\pm\infty$. It is best described as the completion of ${\mathcal{D}}(\mathbb{R})$ equipped with the $\sup$-norm, which will have its importance in the sequel. This property is indicated by $C_{0}(\mathbb{R})=\overline{({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{\sup})}$. The concept is also valid for $L_{1}(\mathbb{R})$, which can be described as $L_{1}(\mathbb{R})=\overline{({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{L_{1}})}$, where the $L_{1}$-norm is defined by (2) with $f\in{\mathcal{D}}(\mathbb{R})$ and the integral being classical—in the sense of Riemann. This completion property applies to $L_{p}(\mathbb{R})=\overline{({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{L_{p}})}$ with $p\in[1,\infty)$ as well [4, Proposition 8.17, p. 254], but not for $p=\infty$, which explains the importance of the space $C_{0}(\mathbb{R})$, which is distinct from $L_{\infty}(\mathbb{R})$.

In order to properly identify $L_{\infty}(\mathbb{R})$ as a subspace of ${\mathcal{D}}^{\prime}(\mathbb{R})$, we shall exploit the property that the $L_{\infty}$-norm is the dual of the $L_{1}$-norm. We therefore choose to define the $L_{\infty}$-norm as

where the central part of (7) takes advantage of the denseness⁵⁵5This means that, for any $f\in L_{1}(\mathbb{R})$ and any $\epsilon>0$, there exists a function $\varphi_{\epsilon}\in{\mathcal{D}}(\mathbb{R})$ such that $\|f-\varphi_{\epsilon}\|_{L_{1}}<\epsilon$. It is a direct consequence of $L_{1}(\mathbb{R})=\overline{({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{L_{1}})}$. of ${\mathcal{D}}(\mathbb{R})$ in $L_{1}(\mathbb{R})$. This yields a definition that is valid not only for (measurable) functions, but also for all $f\in{\mathcal{D}}^{\prime}(\mathbb{R})$. Consequently, we can redefine our target space as

which is readily identified as the topological dual of $L_{1}(\mathbb{R})$; that is, $L_{\infty}(\mathbb{R})=\big{(}L_{1}(\mathbb{R})\big{)}^{\prime}$ due to the dual specification of the $L_{\infty}$-norm given by the right-hand side of (7).

While (8) defines $L_{\infty}(\mathbb{R})$ as a subspace of ${\mathcal{D}}^{\prime}(\mathbb{R})$, we can also identify its elements as (bounded) measurable functions $f:\mathbb{R}\to\mathbb{R}$ via the classical association

where the right-hand side of (9) is a standard Lebesgue integral. Now, the main difference between the $\sup$-norm and the $L_{\infty}$-norm is that, for $f\in L_{\infty}(\mathbb{R})$ (identified as a function), the inequality $|f(t)|\leq\|f\|_{L_{\infty}}$ holds for almost every $t\in\mathbb{R}$. This means that it holds over the whole real line except, possibly, on a set of measure zero. This is often indicated as $\|f\|_{\infty}=\operatorname*{ess\,sup}_{t\in\mathbb{R}}|f(t)|$, using the notion of essential supremum. In other words, the $L_{\infty}$-norm is more permissive than the $\sup$-norm with $\|f\|_{\infty}\leq\|f\|_{\sup}$. However, the two norms are equal whenever the function $f$ is continuous, which translates into the isometric inclusion $C_{\rm 0}(\mathbb{R})\xhookrightarrow{\mbox{\tiny\rm iso.}}C_{\rm b}(\mathbb{R})\xhookrightarrow{\mbox{\tiny\rm iso.}}L_{\infty}(\mathbb{R})$.

Remarkably, the combination of the two latter function spaces enables us to formulate a first Banach extension of the classical statement on BIBO stability. To that end, we shall restrict the distributional framework covered by Theorem 1 to the case where the impulse response $h$ is identifiable as a measurable function (i.e., $h\in L_{1,{\rm loc}}(\mathbb{R})\subset{\mathcal{D}}^{\prime}(\mathbb{R})$). The linear functional on the right-hand side of (6) then has an explicit integral description given by (1) with $f=\varphi\in{\mathcal{D}}(\mathbb{R})$. Within this class of convolution operators, we now identify the ones whose domain can be extended to $L_{\infty}(\mathbb{R})$.

The proof of this result is deferred to Section IV (see Item 2) and the final statement of Theorem 4.

It is of interest to compare Proposition 1 and Theorem 2 because they address the problem of stability from different but complementary perspectives. Proposition 1 is focused primarily on the well-posedness of the convolution integral (1) for $f\in L_{\infty}(\mathbb{R})$. It can be paraphrased as: “Let $h$ be a measurable (and locally integrable) function. Then, the Lebesgue integral (1) defines a convolution operator that is BIBO-stable if and only if $h\in L_{1}(\mathbb{R})$.” By contrast, Theorem 2 considers the complete family of “classical” convolution operators ${\mathrm{T}}_{h}:{\mathcal{D}}(\mathbb{R})\to C(\mathbb{R})$ with $h\in L_{1,\rm loc}(\mathbb{R})$ and precisely identifies the subset of operators that have a continuous extension from $L_{\infty}(\mathbb{R})\to C_{\rm b}(\mathbb{R})$. Since $C_{\rm b}(\mathbb{R})$ is isometrically embedded in $L_{\infty}(\mathbb{R})$, this is more informative than Proposition 1 because it also tells us that $(h\ast f)(t)$ is a continuous function of $t\in\mathbb{R}$. In that respect, we note that the requirement that the convolution of any bounded function $f$ be continuous excludes the use of the identity operator with $h=\delta$ at the onset, even if we extend the framework to $h\in{\mathcal{D}}^{\prime}(\mathbb{R})$.

To obtain a more permissive characterization of BIBO stability, we need to extend the range of the operator from $C_{\rm b}(\mathbb{R})$ to $L_{\infty}(\mathbb{R})$, which should then also translate into a corresponding enlargement of the class of admissible impulse responses. We shall delineate the latter in a way that parallels our definition of $L_{\infty}(\mathbb{R})$, with the roles of the $L_{1}$- and $\sup$- (or $L_{\infty}$-) norms being interchanged. To that end, we first define the ${\mathcal{M}}$-norm as

This then yields the Banach space

which also happens to be the space of bounded Radon measures⁶⁶6We adhere with Bourbaki’s nomenclature to distinguish the two complementary interpretations of a measure: either as a continuous linear functional on ${\mathcal{D}}(\mathbb{R})$ (Radon measure), or as a set-theoretic additive rule that associates a real number to any Borel set of $\mathbb{R}$ (signed Borel measure) [2, 1]. on $C_{0}(\mathbb{R})$. In other words, ${\mathcal{M}}(\mathbb{R})$ is the topological dual of $C_{0}(\mathbb{R})$. Moreover, we can invoke the Riesz-Markov theorem to identify ${\mathcal{M}}(\mathbb{R})=\big{(}C_{0}(\mathbb{R})\big{)}^{\prime}$ with the space of bounded signed Borel measures on $\mathbb{R}$ [15]. Concretely, this means that any $h\in{\mathcal{M}}(\mathbb{R})$ is associated with a unique Borel measure $\mu_{h}$, which then gives a concrete definition of the linear functional

for any measurable function $f$, while the total-variation norm of the measure $\mu_{h}$ is given by $\|\mu_{h}\|_{\rm TV}\stackrel{{\scriptstyle\vartriangle}}{{=}}\int_{\mathbb{R}}{\rm d}|\mu_{h}|=\|h\|_{{\mathcal{M}}}$ (see Section IV-C). The main point for us is that ${\mathcal{M}}(\mathbb{R})$ is a superset of $L_{1}(\mathbb{R})$, with $\|f\|_{{\mathcal{M}}}=\|f\|_{L_{1}}$ for all $f\in L_{1}(\mathbb{R})$. Moreover, we have that $\delta(\cdot-t_{0})\in{\mathcal{M}}(\mathbb{R})$ for any $t_{0}\in\mathbb{R}$ with $\|\delta(\cdot-t_{0})\|_{{\mathcal{M}}}=1$, as can be readily inferred from (10) by considering a non-negative test function that achieves its maximum $\varphi(t_{0})=1$ at $t=t_{0}$.

For the cases where the impulse response $h\in{\mathcal{M}}(\mathbb{R})$ is not an $L_{1}$ function, we extend our definition of the original (Lebesgue) convolution integral as

which is the same as (1) when we can write ${\rm d}\mu_{h}(\tau)=h(\tau){\rm d}\tau$, which happens when the corresponding measure $\mu_{h}$ is absolutely continuous⁷⁷7Another way to put it is that $h$ is the Radon-Nikodym derivative of $\mu_{h}$. with respect to the Lebesgue measure. A standard manipulation then yields that

which is the basis for the direct (easy) part of Theorem 3, where the complete class of BIBO-stable systems is identified, including the identity operator.

This result, which is also valid in dimensions higher than $1$, is known in harmonic analysis [6, p. 140 Corollary 2.5.9], [18] but much less so in engineering circles. It can be traced back to an early paper by Hörmander that provides a comprehensive treatment of convolution operators on $L_{p}$ spaces [7]. The reminder of the paper is devoted to the proof of the two theorems on BIBO stability and of some interesting variants (see Theorem 4). To that end, we shall rely on Schwartz’ powerful distributional formalism which, as we shall see, allows for a rather soft derivation, once the prerequisites have been laid out.

The most general form of a convolution operator backed by Schwartz’ kernel theorem (see Theorem 1) is ${\mathrm{T}}_{h}:{\mathcal{D}}(\mathbb{R})\to C(\mathbb{R})\xhookrightarrow{}{\mathcal{D}}^{\prime}(\mathbb{R})$ with $h\in{\mathcal{D}}^{\prime}(\mathbb{R})$, where ${\mathrm{T}}_{h}\{\varphi\}$ is defined by (6) for any $\varphi\in{\mathcal{D}}(\mathbb{R})$. The two complementary ingredients at play there are: (i) the restriction of the domain to ${\mathcal{D}}(\mathbb{R})$—the “nicest” class of functions in terms of smoothness and decay—and (ii) the extension of the range to ${\mathcal{D}}^{\prime}(\mathbb{R})$, which can accommodate an arbitrary degree of growth (polynomial, or even exponential) at infinity. In other words, the theoretical framework is such that it can deal with the very worst scenarios, including unstable differential systems whose impulse response is exponentially increasing.

Then, depending on the smoothness and decay properties of $h$, it is usually possible to extend the domain of ${\mathrm{T}}_{h}$ to some Banach space ${\mathcal{X}}\supseteq{\mathcal{D}}(\mathbb{R})$ that is continuously embedded in ${\mathcal{D}}^{\prime}(\mathbb{R})$, which is denoted by ${\mathcal{X}}\xhookrightarrow{}{\mathcal{D}}^{\prime}(\mathbb{R})$. For this to be feasible, we require that $\|\cdot\|_{\mathcal{X}}$ be a valid norm on ${\mathcal{D}}(\mathbb{R})$ and that ${\mathcal{D}}(\mathbb{R})$ be dense in ${\mathcal{X}}$, which is equivalent to ${\mathcal{X}}=\overline{({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{{\mathcal{X}}})}$. In other words, ${\mathcal{X}}$ is the completion of ${\mathcal{D}}(\mathbb{R})$ equipped with the $\|\cdot\|_{{\mathcal{X}}}$-norm.

We start by recalling the definition of the norm of a bounded operator.

A direct consequence of Definition 2 is that a bounded operator ${\mathrm{T}}:{\mathcal{X}}\to{\mathcal{Y}}$ continuously maps ${\mathcal{X}}$ into ${\mathcal{Y}}$, as indicated by ${\mathrm{T}}:{\mathcal{X}}\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{Y}}$.

Theorem 2 then describes a functional mechanism that allows us to extend an operator initially defined on ${\mathcal{D}}(\mathbb{R})$. It is a particularization of a fundamental extension theorem in the theory of Banach spaces [14, Theorem I.7, p. 9].

Since a convolution operator ${\mathrm{T}}_{h}:{\mathcal{D}}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{D}}^{\prime}(\mathbb{R})$ is uniquely characterized by its impulse response $h\in{\mathcal{D}}^{\prime}(\mathbb{R})$, the same holds true for its extension ${\mathrm{T}}_{h}:{\mathcal{X}}\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{Y}}$, which justifies the use of the same symbol. Rather than defining ${\mathrm{T}}_{h}\{f\}=h\ast f$ through a Lebesgue integral as in (1) or (13), we can therefore rely on (6) and define our extended convolution operator ${\mathrm{T}}_{h}:{\mathcal{X}}\to{\mathcal{Y}}$ through a limit process. Specifically, we pick a Cauchy sequence $(\varphi_{n})$ in $({\mathcal{D}}(\mathbb{R}),\|\cdot\|_{{\mathcal{X}}})$ such that $\lim_{n\to\infty}\varphi_{n}=f\in{\mathcal{X}}$. Then, the sequence of functions $(g_{n}=h\ast\varphi_{n})$ with

is Cauchy in ${\mathcal{Y}}$ and converges to a limit $g=\lim_{n\to\infty}(h\ast\varphi_{n})\in{\mathcal{Y}}$, independently of the choice of the $\varphi_{n}$ since the space ${\mathcal{Y}}$ is complete. We now recapitulate this process in the form of a definition.

Also important for our purpose is the adjoint operator ${\mathrm{T}}^{\ast}:{\mathcal{Y}}^{\prime}\to{\mathcal{X}}^{\prime}$, which is the unique linear operator such that

for any $g\in{\mathcal{Y}}^{\prime}$ and $f\in{\mathcal{X}}^{\prime}$, where the spaces ${\mathcal{X}}^{\prime}$ and ${\mathcal{Y}}^{\prime}$ are the duals of the topological vector spaces ${\mathcal{X}}$ and ${\mathcal{Y}}$, respectively. If ${\mathrm{T}}:{\mathcal{X}}\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{Y}}$ is bounded with operator norm $\|{\mathrm{T}}\|$, then the adjoint ${\mathrm{T}}^{\ast}:{\mathcal{Y}}^{\prime}\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{X}}^{\prime}$ is bounded with $\|{\mathrm{T}}^{\ast}\|=\|{\mathrm{T}}\|$. In particular, the adjoint of the convolution operator ${\mathrm{T}}_{h}:{\mathcal{D}}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{D}}^{\prime}(\mathbb{R})$ is ${\mathrm{T}}_{h^{\vee}}:{\mathcal{D}}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{D}}^{\prime}(\mathbb{R})$, where $h^{\vee}$ is the time-reversed impulse response such that $\langle h^{\vee},\varphi\rangle=\langle h,\varphi^{\vee}\rangle$, where $\varphi^{\vee}(t)\stackrel{{\scriptstyle\vartriangle}}{{=}}\varphi(-t)$.

We now briefly show how we make use of these two mechanisms to specify the continuous extension ${\mathrm{T}}_{h}:L_{\infty}(\mathbb{R})\to L_{\infty}(\mathbb{R})$ with $h\in{\mathcal{M}}(\mathbb{R})$ (or, $h\in L_{1}(\mathbb{R})$) that is implicitly referred to in Theorems 2 and 3. The enabling ingredient there is the continuity bound $\|h^{\vee}\ast\varphi\|_{L_{1}}\leq\|h^{\vee}\|_{{\mathcal{M}}}\|\varphi\|_{L_{1}}$ (see proof of Theorem 4, Item 4), which also yields $h^{\vee}\ast\varphi\in L_{1}(\mathbb{R})$ for all $\varphi\in{\mathcal{D}}(\mathbb{R})$. We then apply Definition 3 to specify the unique extension ${\mathrm{T}}_{h^{\vee}}:L_{1}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}L_{1}(\mathbb{R})$. An important point for our argumentation is that this (pre-adjoint) convolution operator also has a concrete implementation as

which is supported by the same continuity bound with $\varphi$ now ranging over $L_{1}(\mathbb{R})$ instead of the smaller space ${\mathcal{D}}(\mathbb{R})$. The existence and uniqueness of ${\mathrm{T}}_{h^{\vee}}:L_{1}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}L_{1}(\mathbb{R})$ then guarantees the existence and unicity of the adjoint ${\mathrm{T}}^{\ast}_{h^{\vee}}:L_{\infty}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}L_{\infty}(\mathbb{R})$. To show that ${\mathrm{T}}^{\ast}_{h^{\vee}}={\mathrm{T}}_{h}$, we use the explicit representation of ${\mathrm{T}}_{h^{\vee}}$ given by (19) with $h^{\vee}\in{\mathcal{M}}(\mathbb{R})$ and invoke Fubini’s theorem to justify the interchange of integrals in

for any $f\in L_{1}(\mathbb{R})$ and $g\in L_{\infty}(\mathbb{R})$. This proves that the original convolution operator defined by (13) coincides with the adjoint of ${\mathrm{T}}_{h^{\vee}}:L_{1}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}L_{1}(\mathbb{R})$, which is also consistent with the property $h=(h^{\vee})^{\vee}$. Since ${\mathcal{D}}(\mathbb{R})\subset L_{\infty}(\mathbb{R})$, we can therefore uniquely identify ${\mathrm{T}}_{h}:L_{\infty}(\mathbb{R})\xrightarrow{\mbox{\tiny\ \rm c. }}L_{\infty}(\mathbb{R})$ as the extension of ${\mathrm{T}}_{h}:{\mathcal{D}}(\mathbb{R})\to{\mathcal{D}}^{\prime}(\mathbb{R})$ that preserves the adjoint relation ${\mathrm{T}}^{\ast}_{h^{\vee}}={\mathrm{T}}_{h}$.

The Banach spaces of interest for us are ${\mathcal{X}}=C_{0}(\mathbb{R}),L_{p}(\mathbb{R})$ and ${\mathcal{Y}}=C_{0}(\mathbb{R}),C_{\rm b}(\mathbb{R}),L_{p}(\mathbb{R})$ with $p\geq 1$.