Limits of interpolatory processes

Abstract:

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 \[ \int P_{\epsilon }(x) e^{i \nu _k x} \phi _{\epsilon }(x)dx=f_{\epsilon ,k}, \quad k=1, \ldots , N,\] 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.