Riesz bases formed by exponentials and divided differences (Q2783035)

The authors study divided differences (DDs) of exponentials \(e^{i\lambda_kt}\) in the case where the distance between some points \(\lambda_k\) may be arbitrarily small. D.~Ullrich considered sets \(\Lambda\) of the form \(\Lambda=\cup_{n\in\mathbb Z} \Lambda^{(n)}\), where the subsets \(\Lambda^{(n)}\) consist of equal number of real points \(\lambda_1^{(n)},\dots,\lambda_N^{(n)}\) close to \(n\), i.e., \(|\lambda_j^{(n)}-n|< \varepsilon\) for all \(j\) and \(n\). He proved that, for sufficiently small \(\varepsilon > 0\), the DDs constructed from the subsets \(\Lambda^{(n)}\) form a Riesz basis in \(L^2(0,2\pi N)\). The authors generalize the Ullrich's result in several directions: the set \(\Lambda\) is allowed to be more complicated, and the subsets \(\Lambda^{(n)}\) are allowed to contain an arbitrary number of points, which may fail to be ``very'' close to each other (or even to some integer). Multiple points corresponding to functions of the form \(t^me^{i\lambda t}\) are also considered. Actually, a full description of the Riesz bases of exponential DDs is given. It is shown that the DDs for points \(\lambda_k,\dots,\lambda_N\) lying in a fixed ball form a ``uniform basis'', i.e., that the basis constants do not depend on the positions of the \(\lambda_j\) in the ball. Moreover, the DDs depend on the \(\lambda_j\) analytically. Thus, this family is a natural basis in the situation where the exponentials \(e^{i\lambda t}\) with \(\lambda\in\Lambda\) fail to form even a uniformly minimal family. Application to an observation problem, refining the results of V.~Komornik and P.~Loreti, is given.