Some analytical considerations on two-scale relations J.-P. Antoine and F. Bagarello 1, 2 Institut de Physique Th´ eorique, Universit´ e Catholique de Louvain B - 1348 - Louvain-la-Neuve, Belgium E-mail: Antoine@fyma.ucl.ac.be ; bagarello@ipamat.cres.it Abstract Scaling functions that generate a multiresolution analysis (MRA) satisfy, among other conditions, the so-called ‘two-scale relation’ (TSR). In this paper we discuss a number of properties that follow from the TSR alone, independently of any MRA: position of zeros (mainly for continuous scaling functions), existence theorems (using fixed point and eigenvalue arguments) and orthogonality relation between integer translates. UCL-IPT-94-04 February 1994 1 Work supported by ‘Fondazione Angelo Della Riccia’ 2 Permanent address : Dipartimento di Matematica ed Applicazioni, Fac. Ingegneria, Universit` a di Palermo, I - 90128 Palermo, Italy