I'm reading few papers about signal processing and the following theorem is stated.
A function $f(x,y)$ steers when it can be written as $$ f^{\theta}(x,y) = \sum_{j=1}^M k_j(\theta)f^{\theta_j}(x,y) $$
And the goal would be finding which functions admit such decomposition, and how many terms $M$ are required.
Theorem 1 : The steering condition holds for functions expandable as $$ f(r,\phi) = \sum_{n=-N}^N a_n(r)e^{in\phi} $$ if and only if the interpolation functions $k_j(\theta)$ are solutions of $$ \begin{pmatrix}1 \\e^{i\theta} \\ \vdots \\ e^{iN\theta} \end{pmatrix} = \begin{pmatrix} 1 & 1 & \ldots & 1 \\ e^{i\theta_1} & e^{i\theta_2} & \ldots & e^{i\theta_M} & \\ \vdots & \vdots & & \vdots \\ e^{iN\theta_1} & e^{iN\theta_2} & \ldots & e^{iN\theta_M} \end{pmatrix} \begin{pmatrix} k_1(\theta) \\ k_2(\theta) \\ \vdots \\ k_M(\theta) \end{pmatrix} $$
Not sure about the proof of this theorem though, it's just few lines and basically it seems to me $f(r,\phi)$ substitutes both $f^{\theta}$ and $f^{\theta_j}$ in the steerable condition, we pick each component $m$ and from there we derive the linear system, is this the idea of the proof of this theorem?
The other theorem is the following
Theorem 2: Let $T$ be the number of nonzero coefficients $a_n(r)$ for a function $f(r,\phi)$ expandable in the form above. Then, the minimum number of basis functions sufficient to steer $f(r,\phi)$ by $$ f^{\theta}(r,\phi) = \sum_{j=1}^{M} k_j(\theta)g_j(r,\phi) $$ is $T$, i.e. $M$ must be $\geq T$.
The proof of this theorem is similar to the former in the initial part, i.e. we substitute the expansion above to both $f^{\theta}$ and $g_j$, we later project into $e^{im\theta}$ for $0 \leq m \leq N$, and divide both sides by $a_m(r)$ which provides
$$ e^{im\theta} = \sum_{j=1}^M k_j(\theta)c_{jm}(r). $$
From here not entirely sure how an equation of the form
$$ I = CKC^T $$
is derived. The paper is enter link description here, and again the questions are if you can fill the bits of the proof of the two theorems.
Update (notation clarification) : From the paper
Let $(\ldots)^{\theta}$ represent the rotation operator such that for any function $f(x,y)$, $f^{\theta}(x,y)$ is $f(x,y)$ rotated through an angle $\theta$ about the origin.