Scale-dependent functions in statistical hydrodynamics: a functional analysis point of view

Most of dynamic systems which exhibit chaotic behavior are also known to posses self-similarity and manifest strong fluctuations of all possible scales.The meaning of this terms is not always same. In present note we make an attempt to formulate the problem in the framework of functional analysis. The statistical hydrodynamics is taken as a vivid physical example. The links to wavelet analysis, multi-fractal formalism and Nottale’s scale relativity are presented.