Dense analytic subspaces in fractal L 2 -spaces
Abstract
We consider self-similar measures
μ
with support in the interval
0≤x≤1
which have the analytic functions
{
e
i2πnx
:n=0,1,2,...}
span a dense subspace in
L
2
(μ)
. Depending on the fractal dimension of
μ
, we identify subsets
P⊂
N
0
={0,1,2,...}
such that the functions
{
e
n
:n∈P}
form an orthonormal basis for
L
2
(μ)
. We also give a higher-dimensional affine construction leading to self-similar measures
μ
with support in
R
ν
. It is obtained from a given expansive
ν
-by-
ν
matrix and a finite set of translation vectors, and we show that the corresponding
L
2
(μ)
has an orthonormal basis of exponentials
e
i2πλ⋅x
, indexed by vectors
λ
in
R
ν
, provided certain geometric conditions (involving the Ruelle transfer operator) hold for the affine system.