On the Structure of Self-similar Systems: A Hilbert Space Approach

This paper investigates the structural properties of linear self-similar systems, using an invariant subspace approach. The self-similar property is interpreted in terms of invariance of the corresponding transfer function space to a given transformation in a Hilbert space, in a same way as the time invariance property for linear systems is related to the shift-invariance of the Hardy spaces. The transformation in question is exactly that defining the de Branges homogeneous spaces. We show that any de Branges homogeneous space of order \(\nu \geqslant - \frac{1} {2}\) belongs to the Paley-Wiener space so that each element of such a space may be viewed as the transfer function of some linear self-similar system of parameter v. The explicit form of the corresponding impulse response, which is shown to be described by a hyperbolic partial differential equation, is given. Finally, we emphasize on the infinite dimension nature of self-similar systems through an abstract state space description.

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