Multiscale version of the Logvinenko-Sereda Theorem
Logvinenko-Sereda 定理的多尺度版本
基本信息
- 批准号:280969390
- 负责人:
- 金额:--
- 依托单位:
- 依托单位国家:德国
- 项目类别:Research Grants
- 财政年份:2015
- 资助国家:德国
- 起止时间:2014-12-31 至 2018-12-31
- 项目状态:已结题
- 来源:
- 关键词:
项目摘要
The aim of this project is to prove a multiscale version of the Logvinenko-Sereda Theorem.The classical Logvinenko-Sereda Theorem belongs to the realm of Harmonic or Fourier Analysis and asserts that the restriction of an $L^p$ function on the real lineto a thick subset has comparable $L^p$-norm to the function on the whole axis, provided that its Fourier transform is supported on a compact interval. Only the size of the interval enters in the estimate, not its position. The thicknessof the restriction set enters in the constant as well. Kovrijkine extended the result to the case where the Fourier transform has support in a union of a finite number of intervals of the same length. Again, only the number and the size of the intervals enters the estimate, not their position.Recent scale-free unique continuation estimates or uncertainty principles for eigenfunctions and spectral projections of Schrödinger operator suggest the validity of a multiscale version of the Logvinenko-Sereda Theorem.Here the functions are considered on intervals of length $L$, where $L$ ranges over the positive reals.While the intended estimate appears at first sight simpler than in the case of the whole axis, one has the additional taskto control effectively the dependence of the estimate on the additional size parameter $L$. Ideally, one would like to show that the estimateshold uniformly in $L$. While the conjectured bound is a Harmonic Analysis result, it immediately triggers consequencesin the theory of Inverse Problems, in particular under appropriate sparsity assumptions. Thus it can be seen as an continuum relative of compressed sensing and sparse recovery. Moreover, the conjectured estimateshave applications in the spectral theory of Schrödinger operators and in the control theory of the heat equation.A multidimensional extension of this estimate will have even wider relevance.
经典的Logvinenko-Sereda定理属于调和分析或傅立叶分析的范畴,它断言,如果一个函数的傅立叶变换在紧区间上得到支持,那么它在真实的直线上的厚子集上的L ^p$-范数与在整个轴上的函数的L^p$-范数相当。只有区间的大小进入估计,而不是它的位置。限制集的厚度也进入常数。Kovrijkine将结果扩展到傅立叶变换在有限个相同长度的区间的联合中具有支持的情况。再次,只有区间的数目和大小进入估计,而不是他们的位置。最近的无标度唯一连续估计或不确定性原理的本征函数和谱投影的薛定谔算子建议的多尺度版本的Logvinenko-Sereda定理的有效性。这里的功能被认为是在区间长度$L$,其中L在正实数范围内变化。虽然预期的估计乍一看比整个轴的情况简单,但我们有额外的任务来有效地控制估计对附加尺寸参数L的依赖性。理想情况下,人们希望证明估计值在$L$中是一致的。虽然约束边界是调和分析的结果,但它会立即触发反问题理论中的结果,特别是在适当的稀疏性假设下。因此,它可以被看作是压缩感知和稀疏恢复的连续体。此外,该估计在薛定谔算子的谱理论和热方程的控制理论中也有应用,其多维推广将具有更广泛的意义.
项目成果
期刊论文数量(3)
专著数量(0)
科研奖励数量(0)
会议论文数量(0)
专利数量(0)
Sharp estimates and homogenization of the control cost of the heat equation on large domains
- DOI:10.1051/cocv/2019058
- 发表时间:2018-10
- 期刊:
- 影响因子:0
- 作者:Ivica Naki'c;Matthias Täufer;Martin Tautenhahn;I. Veselić
- 通讯作者:Ivica Naki'c;Matthias Täufer;Martin Tautenhahn;I. Veselić
Null-controllability and control cost estimates for the heat equation on unbounded and large bounded domains
- DOI:10.1007/978-3-030-35898-3_5
- 发表时间:2018-10
- 期刊:
- 影响因子:0
- 作者:Michela Egidi;Ivica Naki'c;Albrecht Seelmann;Matthias Taufer;Martin Tautenhahn;I. Veselić
- 通讯作者:Michela Egidi;Ivica Naki'c;Albrecht Seelmann;Matthias Taufer;Martin Tautenhahn;I. Veselić
Scale-free Unique Continuation Estimates and Logvinenko–Sereda Theorems on the Torus
环面上的无标度唯一连续估计和 LogvinenkoâSereda 定理
- DOI:10.1007/s00023-020-00957-7
- 发表时间:
- 期刊:
- 影响因子:0
- 作者:M. Egidi;I. Veselić
- 通讯作者:I. Veselić
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Professor Dr. Ivan Veselic其他文献
Professor Dr. Ivan Veselic的其他文献
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394221243 - 财政年份:2018
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27091790 - 财政年份:2006
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Spectral properties of random Schroedinger operators and random operators on manifolds and graphs
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5423391 - 财政年份:2004
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Analysis of spectral properties of solid-state Schrödinger operators.
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5371487 - 财政年份:2002
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441959487 - 财政年份:
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