Compactly supported solutions of two-scale difference equations (Q1307195)

A two-scale difference equation is a functional equation of the form \[ \varphi\left( {t\over 2}\right)= \sum^n_{\nu=0} C_\nu\varphi(t-\nu), \tag{E} \] where \(C_\nu\) are given real or complex constants with \(C_0C_n\neq 0\) and \(n\geq 1.\) A function \(\varphi\) satisfying (E) for all real \(t\) is called refinable. The authors consider Lebesgue integrable, compactly supported solutions of (E) and investigate the relations between translates of these solutions. A detailed study of corresponding invariant subspaces leads to new observations concerning the factorization of the refinement mask and certain spectral properties of the corresponding coefficient matrices. In particular new necessary conditions for the existence of integrable compactly supported solutions are derived.