Limits of interpolatory processes

Given $N$ distinct real numbers $\nu_1, \ldots, \nu_N$ and a positive approximation of the identity $\phi_{\epsilon}$, which converges weakly to the Dirac delta measure as $\epsilon$goes to zero, we investigate the polynomials $P_{\epsilon}(x)= \sum c_{\epsilon , j} e^{-i \nu_j x}$ which solve the interpolation problem

with prescribed data $f_{\epsilon,1}, \dots, f_{\epsilon,N}$. More specifically, we are interested in the behavior of $P_{\epsilon}(x)$ when the data is of the form $f_{\epsilon, k}=\int f(x) e^{i \nu_k x} \phi_{\epsilon}(x)dx$ for some prescribed function $f$. One of our results asserts that if $f$ is sufficiently nice and $\phi_{\epsilon}$ has sufficiently well-behaved moments, then $P_{\epsilon}$ converges to a limit $P$ which can be completely characterized. As an application we identify the limits of certain fundamental interpolatory splines whose knot set is $\mathbb{Z} \setminus \mathcal{N}$, where $\mathcal{N}$ is an arbitrary finite subset of the integer lattice $\mathbb{Z}$, as their degree goes to infinity.