I'm reading through enter link description here this paper. And I'm trying to understand the proof of the proposition 1, which states:
The transformation defined by $$ y[n] = \int_{0}^{2\pi} dx h(n\Delta_x - x)f(x) $$ is shiftable if and only if there's a set of interpolation functions, $b_n(x_0)$ that satisfy the matrix equation $$ \begin{pmatrix} e^{jx_0k_0} \\ e^{jx_0k_1} \\ \vdots \\ e^{jx_0k_{M-1}} \end{pmatrix} = \begin{pmatrix} 1 & e^{j\Delta_x k_0} & e^{j 2 \Delta_x k_0} & & e^{j(N-1)\Delta_x k_0} \\ 1 & e^{j\Delta_x k_1} & e^{j 2 \Delta_x k_1} & \ldots & e^{j(N-1)\Delta_x k_1} \\ & \vdots & & \\ 1 & e^{j\Delta_x k_{M-1}} & e^{j 2 \Delta_x k_{M-1}} & & e^{j(N-1)\Delta_x k_{M-1}} \end{pmatrix} \begin{pmatrix} b_0(x_0) \\ b_1(x_0) \\ \vdots \\ b_{N-1}(x_0) \\ \end{pmatrix} \;, \forall x_0 $$
The paper proves such proposition. And I understand the proof. However the form of the linear system above doesn't seem to me depends from the $h$, and therefore to me it seems that for any $h$ we have a shiftable transformation.
PS. Shiftability means that for all $x_0$ we have
$$ \int_{0}^{2\pi} dx h(x_0 - x)f(x) = \sum_{j=0}^{N-1} b_j(x_0) y[n] $$
Namely convolution can be expressed as interpolation of $N$ samples. Notation in general of the paper is not entirely clear, for example Fourier series of $h(x)$ is denoted by $H(k)$, with a different variable, while to me the variable should be the same.
The statement of the proposition continues after presenting the linear system. It continues by saying: "where $\{k_0, k_1, \dots, k_{M-1}\}$ is the set of frequencies for which $H(k)$, the Fourier series of $h(x)$, is nonzero". I guess that gives your dependence on $h$.