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Riesz bases formed by exponentials and divided differences. (English. Russian original) Zbl 0999.42018

St. Petersbg. Math. J. 13, No. 3, 339-351 (2002); translation from Algebra Anal. 13, No. 3, 1-17 (2002).
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.

MSC:

42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
42B05 Fourier series and coefficients in several variables
93C20 Control/observation systems governed by partial differential equations
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