Ingham-type inequalities and Riesz bases of divided differences

Sergei Avdonin; William Moran

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 4, page 803-820
  • ISSN: 1641-876X

Abstract

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We study linear combinations of exponentials e^{iλ_nt} , λ_n ∈ Λ in the case where the distance between some points λ_n tends to zero. We suppose that the sequence Λ 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 Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{iλ_nt}} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from {e^{iλ_nt}} a suitable collection of functions e^{iλ_nt}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.

How to cite

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Avdonin, Sergei, and Moran, William. "Ingham-type inequalities and Riesz bases of divided differences." International Journal of Applied Mathematics and Computer Science 11.4 (2001): 803-820. <http://eudml.org/doc/207532>.

@article{Avdonin2001,
abstract = {We study linear combinations of exponentials e^\{iλ\_nt\} , λ\_n ∈ Λ in the case where the distance between some points λ\_n tends to zero. We suppose that the sequence Λ 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 Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to \{e^\{iλ\_nt\}\} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from \{e^\{iλ\_nt\}\} a suitable collection of functions e^\{iλ\_nt\}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.},
author = {Avdonin, Sergei, Moran, William},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {string equation; divided differences; beamequation; Riesz bases; simultaneous controllability; beam equation},
language = {eng},
number = {4},
pages = {803-820},
title = {Ingham-type inequalities and Riesz bases of divided differences},
url = {http://eudml.org/doc/207532},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Avdonin, Sergei
AU - Moran, William
TI - Ingham-type inequalities and Riesz bases of divided differences
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 4
SP - 803
EP - 820
AB - We study linear combinations of exponentials e^{iλ_nt} , λ_n ∈ Λ in the case where the distance between some points λ_n tends to zero. We suppose that the sequence Λ 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 Λ is less than T/(2π), the family of divided differences can be extended to a Riesz basis in L^2 (0,T) by adjoining to {e^{iλ_nt}} a suitable collection of exponentials. Likewise, if the lower uniform density is greater than T/(2π), the family of divided differences can be made into a Riesz basis by removing from {e^{iλ_nt}} a suitable collection of functions e^{iλ_nt}. Applications of these results to problems of simultaneous control of elastic strings and beams are given.
LA - eng
KW - string equation; divided differences; beamequation; Riesz bases; simultaneous controllability; beam equation
UR - http://eudml.org/doc/207532
ER -

References

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