The calculation of tasks related to large complex systems usually calls for the application of the most suitable methods and space representations. This, however, requires a review process which checks whether the calculation transformations satisfy the group conditions. These primarily concern the property compliance of the system’s mathematical model, including the stationarity of its significant behaviour characteristics, such as stability, and the validity of the norms influencing them.

The uniform approach therefore needs to be enforced both in the time and frequency domain and in the continuous and discrete realisations. The adequacy of the time and frequency domains are enabled by the Taylor and Fourier series and the Fourier-Laplace transformations between the Laplace transforms, while the other conditions are enabled by the parametrisation of the Blaschke functions with Möbius transformation and its application as the variables of the Blaschke series. In the geometric Möbius interpretation of the Blaschke function, dilatation and rotation is expressed in this role by time dilatation or shift which represents the changing process manifesting itself in a discrete (e.g. digital) form, in a reversed direction from an observation perspective and toward the prediction future from a control perspective. Accordingly, rotation in the frequency domain performs filtering by frequency, practically the model of holographic imaging.

The Blaschke function establishes a bijective relation (one-to-one correspondence) between the infinitely vast time-space and the dilatation compression and rotation transformations, where the projection can be performed to the circle or the annulus (the Poincaré torus) and the other conical-section-spaces according to the parametrisation of the Blaschke function in the hyperbolic spaces.

Selecting spaces that fit the task enables the use of orthogonal function series by which the singularities (poles) of the process are given by the eigenvalues of the function series matrices (as the case may be, Gram matrices). This way orthogonality is a tool of system identification. Reverting the transformations would show that these poles are the roots of the characteristic equations of the Taylor series, the processes represented in the time domain, or the eigenfrequencies of the system in the frequency domain. Orthogonal functions are usually Laguerre polynomials or similar expressions.

The product elements of the Blaschke series represent the relations of the system singularities by element and their absolute values represent their distance to singularity.

Glossary