Summary: In a series of papers I. Sandberg has (with some justification) criticized the way how first year engineering students are typically introduced to the theory of linear, translation invariant systems. He demonstrates that, strictly speaking, the way how the impulse response is derived, using the so-called sifting property of the Dirac delta “function,” is not conclusive. He provised a “counter-example,” which is a non-zero translation invariant operator \(T\) on \(\mathbf{C}_{b}(\mathbb{R}^{d}))\) which is \(not\) a convolution operator with a bounded measure, the so-called impulse response. This example is not constructive, because it is based on the Theorem of Hahn-Banach and hence on Zorn’s Lemma, but it is nevertheless based on a correct and critical observation.
We will demonstrate that the problem “disappears” if one replaces the non-separable space \(\left(\mathbf{C}_{b}(\mathbb{R}^{d}),||\cdot ||_{\infty}\right)\) by the smaller space \(\left(\mathbf{C}_{0}(\mathbb{R}^{d}),||\cdot ||_{\infty}\right)\) which consists of all the continuous functions which vanishing at infinity, i.e., satisfy \(\lim_{x \to \infty}f(x) = 0\). By proposing a mathematically correct problem description which requires only basic concepts of functional analysis we hope to convince the readers that engineers should not be lead to just ignore the necessity for mathematical correctness, but rather ask whether their intuitive expectations can be turned onto a valid mathematical model by choosing suitable mathematical tools.
The proposed approach has the big advantage of being applicable for general locally compact Abelian (LCA) groups, so the approach is suitable in the spirit of a unified signal analysis (incorporating discrete and continuous, periodic and non-periodic signals). Furthermore it does not require Lebesgue integrals or the Haar measure over such groups.
At the same time it allows to introduce the concept of convolution for bounded measures in a very intuitive way. Despite the possible level of generality we will formulate the main results for the setting of the Euclidean space \(\mathbb{R}^{d}\). However, to the extent convenient for the presentation, we avoid tools which are specific to this setting, such as dilation arguments.
For the entire collection see [Zbl 1378.42001].