Dense Analytic Subspaces In Fractal L²-Spaces

We show that for certain self-similar measures ¯ with support in the interval 0 x 1, the analytic functions \Phi e i2ßnx : n = 0; 1; 2; : : : \Psi contain an orthonormal basis in L 2 (¯). Moreover, we identify subsets P ae N0 = f0; 1; 2; : : : g such that the functions fen : n 2 Pg form an orthonormal basis for L 2 (¯). We also give a higher-dimensional affine construction leading to selfsimilar measures ¯ with support in R , obtained from a given expansive -by- matrix and a finite set of translation vectors. We show that the corresponding L 2 (¯) has an orthonormal basis of exponentials e i2ß\Deltax , indexed by vectors in R , provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system