If you care for multidimensional signals, let us use "shift-invariant" maps. To analyze such systems, Fourier transforms are tools (for engineers) of choice. I often say that Fourier transforms are to convolutions what logarithms are to multiplications: a tool that helps explaining what is less than obvious (diagonalization of the convolution operators), and that can become computationally efficient (FFT), and sometimes a game changer: mp3 and JPEG formats boosted the use of Internet.
Meanwhile, invariance is a very useful concept, and Fourier is a central operator in mathematics, and still subject to many research and open questions (for instance on series convergence). I had the opportunity to hire a skilled mathematics student (Jérôme Gauthier) for a PhD in signal processing on the synthesis in oversampled complex filter banks. I remember that during the overview, as a mere DSP engineer, I asked him about his knowledge on the Fourier transform, and he was (then) pretty confident. Later, he discovered another side of it (by the way, I rediscovered the least-squares parabolic extrapolation recently, which I though I mastered, from the computer science and real-time computing point of view).
So, the interplay between engineers and mathematicians is quite important, because one can shed a different light on how the other deals with it. For instance, research on the mathematics behind invariance is helpful to design novel algorithms. For instance, finding invariant transformations of graphs is of current interest.