TWO - SCALE DIFFERENCE EQUATION : LOCAL AND GLOBAL LINEAR INDEPENDENCE 3 for N

Let be a distribution solution of the two-scale diierence equation (1). First the equivalence of local and global linear independence of the integer translates of is proved and a simple characterization for global linear independence of the integer translates of is given. Second a class of functions in V 1 such that their integer translates are locally or globally linearly independent is found. The objective of this context is to study local and global linear independence of the integer translates of a distribution solution of the two-scale equation. To this end, we introduce some notations and deenitions. Let fc k g N k=0 be a sequence such that c 0 6 = 0; c N 6 = 0 and P N k=0 c k = 2: Let be a unique complex-valued compactly supported distribution to satisfy a two-scale diierence equation (x) = N X k=0 c k (2x ? k) ^ (0) =1; (1) where the Fourier transform b of is deened by b (x) = Z e ?ixx (x)dx: