Resonance or viscosity or damping easily spoils scale-invariance. The resonant frequency of a filter shifts if we stretch the time axis. The difference equations

(5) | ||

(6) |

Another aspect to scale-invariance work is the presence
of ``parasitic'' solutions,
which exist but are not desired.
For example, another solution to *y*_{t} - *y*_{t-2}=0
is the one that oscillates at the Nyquist frequency.

(Viscosity does not necessarily introduce an inherent length and thereby spoil scale-invariance. The approximate frequency independence of sound absorption per wavelength typical in real rocks is a consequence of physical inhomogeneity at all scales. See for example Kjartansson's constant Q viscosity, Q described in IEI. Kjartansson teaches that the decaying solutions are scale-invariant. There is no ``decay time'' for the function .Differential equations of finite order and difference equations of finite order cannot produce damping, yet we know that such damping is important in observations. It is easy to manufacture damping in Fourier space; is used. Presumably, difference equations can make reasonable approximations over a reasonable frequency range.)

4/27/2004