Dense analytic subspaces in fractal L2-spaces

We show that for certain self-similar measures mu with support in the interval 0 less than or equal to x less than or equal to 1, the analytic functions {e(i2 pi nx) : n = 0, 1, 2, ...} contain an orthonormal basis in L-2(mu). Moreover, we identify subsets P subset of N-0 = {0, 1, 2,...} such that the functions {e(n): n is an element of P} form an orthonormal basis for L-2(mu). We also give a higher-dimensional affine construction leading to self-similar measures mu With support in R-nu, obtained from a given expansive nu-by-nu matrix and a finite set of translation vectors. We show that the corresponding L-2(mu) has an orthonormal basis of exponentials e(i2 pi lambda.x), indexed by vectors lambda in R-nu, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.