Courses on linear systems theory describe linear operators which commute with time-shifts as convolution operators or via the transfer function. Whereas composition is realized as convolution in the time-domain it is a simple multiplication operation in the frequency domain. For the continuous variable case (one or multi-dimensional) the Fourier transform is often introduced as an integral transform, which seemingly requires the use of the Lebesgue integral. Nevertheless one makes use of certain divergent integrals in order to describe the inverse Fourier transform or the sifting property of the Dirac Delta thing....

A way out of these difficulties is to work with the space of tempered distributions, which are defined as the continuous dual of the space of rapidely descreasing functions in the sense of L. Schwartz, which is a nuclear Frechet space with a countable system of seminorms.

This being a rather complicated object usually beyond reach for engineers it makes sense to look out for more simple, but still mathematically correct alternatives.

The talk will promote a not yet popular Banach space of test functions, the Segal algebra $S_0(Rd)$ which consists of all $L2$ functions with INTEGRABLE short-time Fourier transform (e.g. with respect to a Gaussian window). The dual space $S_0*(Rd)$ consists of all tempered distribution with bounded spectrogram. Together with the Hilbert space $L2(Rd)$ of square integrable functions they form a so-called Banach Gelfand Triple. In order to explain how it can be used the comparison to the triple of number systems Q in R in C is helpful.

For most applications (aside from the discussion of PDEs) in the domain of engineering this setting provides a solid mathematical justification of the necessary terms and arguments needed for engineering applications. The space of test functions is Fourier invariant and thus also its dual space. The Fourier transform converts pure frequencies into Dirac Delta Distributions. Sampling corresponds to pointwise multiplication of test functions by a Dirac comb. Via the generalized Fourier transform this corresponds to convolution with (another, dilated) Dirac comb, i.e. periodization. Even a so called kernel theorem is possible in this setting, i.e. one can describe operators (very much like matrix representations for finite, discrete signals) via their distributional kernel. Hence we have a setting very reminiscent of the Schwartz setting, but in principle much easier to establish, and requiring less sophisticated concepts, just Banach spaces, there duals and the concept of norm resp. pointwise convergence.