Geodesic
Ingham-Type Inequalities and Riesz Bases of Divided Differences
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 803-820.

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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.
Keywords: simultaneous controllability, string equation, beam equation, Riesz bases, divided differences
Mots-clés : bazy Riesza, sterowanie jednoczesne 
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     author = {Avdonin, S. and Moran, W.},
     title = {Ingham-Type {Inequalities} and {Riesz} {Bases} of {Divided} {Differences}},
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Avdonin, S.; Moran, W. Ingham-Type Inequalities and Riesz Bases of Divided Differences. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 803-820. http://geodesic.mathdoc.fr/item/IJAMCS_2001_11_4_a2/