# Ingham-type inequalities and Riesz bases of divided differences

International Journal of Applied Mathematics and Computer Science (2001)

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

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topAvdonin, 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 -

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## Citations in EuDML Documents

top- S. A. Avdonin, B. P. Belinskiy, L. Pandolfi, Controllability of a Nonhomogeneous String and Ring under Time Dependent Tension
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- Sergei Avdonin, Abdon Choque Rivero, Luz de Teresa, Exact boundary controllability of coupled hyperbolic equations

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