Designing a FIR filter in frequency domain: Paley-Wiener "the other way around". This question is about designing a FIR filter in Paley-Wiener space by specifying its behaviour with respect to a finite set of frequencies.
From the classical Paley-Wiener theorem(s) we get the following conditions sufficient for a function $h(t)$ to be real-valued with compact support in the interval $[0,A]$:
Let $A>0$ and let $\hat{h}\left(z\right):\;\mathbb{C}\to\mathbb{C}$ be a function with the following properties:
1.) $\hat{h}$ is an entire function.
2.) For $x\in\mathbb{R}$ the function $\Re\left(\hat{h}\left(x\right)\right)$ is even.
3.) For $x\in\mathbb{R}$ the function $\Im\left(\hat{h}\left(x\right)\right)$ is odd.
4.) $\exists c<\infty\;:\;\sup_{y>0}\intop_{-\infty}^{\infty}\left|\hat{h}\left(x+i\cdot y\right)\right|^{2}\mathtt{d}x\leq c$
5.) $\exists\tilde{c}<\infty\;\forall z\in\mathbb{C}\;:\;\left|\hat{h}\left(z\right)\right|\leq\tilde{c}\cdot e^{A\left|z\right|}$
Then $h\left(t\right)=\intop_{-\infty}^{\infty}\hat{h}\left(x\right)e^{-itx}\mathtt{d}x$ is a real-valued function with compact support $\mathtt{supp}\,h\subseteq\left[0,A\right]$.
The idea is now to specify the behaviour of $h$ with respect to a finite set of frequencies by requiring $\hat{h}$ to take certain values at those frequencies:
For $j\in\left[0,N-1\right]$, $\;f_{j}\in\mathbb{R}_{+}$ and $k_{j}\in\mathbb{C}$, we require that $\hat{h}\left(f_{j}\right) = k_{j}$.
In order to construct such a $\hat{h}$, a Riesz-basis for the Paley-Wiener space would be helpful.
Unfortunately, most of the Paley-Wiener theory applied to filter design deals with the opposite setting of band-limited functions, where only a real-valued and even Riesz-basis of the subspace of real-valued and even functions of frequency is used in order to receive a IIR filter.
We do however require a Riesz-basis for the whole PW-space. While obviously only its values along the real axis are of practical interest, we still need the theoretical properties of that basis as a set of complex-valued functions of a complex variable so that 1.-5. will hold.
I would appreciate any hints to readings relevant to the setting described here and on how to obtain such a basis.
Thank you!
Edit:
In order not to bloat this question, I will collect any progress I make on this issue in my own answer below.
 A: I came across a quite surprising result by K. M. Dyakonov [1], that relates to the imaginary part of the PW-functions' restrictions to the real line. That would be of interest with respect to the phase-response of the filter to be designed.
The result is as follows:
Let $f\in\mathtt{PW}$ and $\phi=\hat{f}$. Suppose that $f\left(z\right)$ has no zeros for $\Im\left(z\right)>0$. Then the following are equivalent:
i) $$\exists\alpha\in\mathbb{C}\;:\; x\in\mathbb{R}\;\Rightarrow\; f\left(x\right)\in\alpha\mathbb{R}$$
ii) $$\exists A>0\;:\;\mathtt{supp}\phi=\left[-A,\, A\right]$$
  and the following holds:
$$\intop_{-\infty}^{\infty}\, t\left|\phi\left(t\right)\right|^{2}\,\mathtt{d}t=0$$
Note that the support must be actually symmetric and not just contained in such a symmetric interval. The function $\phi$ will be complex-valued in general, for real $\phi$ an immediate consequence is that $\alpha \in \mathbb{R}$. Also it is known from PW-theory that $\phi \in L^2$.
The surprising part is the equivalence ii) $\Rightarrow$
  i).
[1] Konstantin M. Dyakonov: "On the zeros and Fourier transforms of entire functions in the Paley-Wiener space" - DOI: 10.1017/S0305004100074211
