Let $H$ be some function space e.g $L^2[a,b] $ or $L^2(\mathbb{R})$. (Though I am happy to hear about other spaces).
Consider the sequence of operators $D_n:H \to H$ where $D_n(f)(x) := f(nx)$. (So I am assuming that $H$ is a space for which $D_n$ is a continuous linear operator from $H$ to itself.) Are there any theorems which attempt to characterize when $\{D_n(f) \}_{n \in \mathbb{Z}} $ satisfies ANY of the following:
$1.)$ Linearly independent in $H$
$2.)$ Complete in $H$
$3.)$ Basis for $H$
$4.)$ Orthogonal basis for $H$
There are a variety of theorems which make characterizations of this form for other sequences of operators. Some examples of these operators are
1.) $T_{n}(f)(x) = f(x -n )$
2.) $M_n(f)(x) = e^{2\pi i nx}f(x)$
3.) $D_{2^n}f (x) = f(2^n x) $
All the characterizations I am aware of involve the Fourier Transform. Despite the usefulness of Fourier series, I haven't been able to find any similar characterizations for the integer dilations of a function.