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1 International Journal of Applied Mathematics and Computer Science

1 2001

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Ingham-Type Inequalities and Riesz Bases of Divided Differences

Avdonin S., Moran W.

International Journal of Applied Mathematics and Computer Science

2001

Vol. 11, no 4

803-820

We study linear combinations of exponentials e^{i lambda_n t}, lambda_n in Lambda in the case where the distance between some points lambda_n tends to zero. We suppose that the sequence Lambda is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L^2 (0,T). Here we prove that if the upper uniform density of Lambda is less than T/(2 pi), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{i lambda_n t} } a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2 pi), the family of divided differences can be made into a Riesz basis by removing from {e^{i lambda_n t} } a suitable collection of functions e^{i lambda_n t}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.