Multiscale version of the Logvinenko-Sereda Theorem

经典的Logvinenko-Sereda定理属于调和分析或傅里叶分析的范畴，它断言，如果一个函数的傅里叶变换被支持在紧区间上，则实直线上的$L^p$函数对稠子集的限制具有与整个轴上的函数类似的$L^p$-范数。只有区间的大小才会进入估计值，而不是其位置。限制集的厚度也进入常量。Kovrijkine将结果推广到傅里叶变换具有相同长度的有限个区间的并集的情况。同样，只有区间的数目和大小进入估计，而不是它们的位置。最近的无标度唯一连续估计或特征函数的不确定性原理和薛定谔算子的谱投影表明了多尺度版本的Logvinenko-Sereda定理的有效性。这里的函数被考虑在长度为$L$的区间上，其中$L$在正实数上范围。虽然预期的估计乍看起来比整个轴的情况更简单，但人们有额外的任务来有效地控制估计对额外的尺寸参数$L$的依赖。理想情况下，人们希望表明，估计以$L$为单位是一致的。虽然猜想的边界是调和分析的结果，但它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，它可以看作是压缩感知和稀疏恢复的一个连续体。此外，猜想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，这个估计的多维推广将具有更广泛的意义。

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.