The purpose of this talk will be to indicate that abstract mathematical view-points are not only able to obtain a unified view-point on many problems appearing in signal analysis, but also allow to focus on the essential aspects of tools relevant in electrical engineering and other application areas.

The view-point of linear algebra allows to discuss problems such as minimal norm least squares approximation in the context of Hilbert spaces. Additional functional analytic considerations using only Banach spaces and a simplified theory of generalized functions allows to provide a mathematical clean description of situations arising in engineering or theoretical physics. In a way, the attention is shifted away from questions of existence and convergence, to the more lucid algebraic facts that can be used in concrete situations. Just think of the equation $ exp(2 pi i) == 1$, which is powerful and easily used, while it might appear to be a strange object, taking an irrational number and applying a complex, irrational exponent to it.

Having established clear rules within this theory, including Dirac point measures of comb-distributions many facts easily verified in the linear algebra context, or tested using MATLAB resp. OCTAVE, can be transfered mathematically correctly into the continuous setting.