Multiscale version of the Logvinenko-Sereda Theorem

• Applicant: Professor Dr. Ivan Veselic
• Subject Area: Mathematics
• Term: from 2015 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 280969390

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.

• DFG Programme: Research Grants
• International Connection: Croatia, United Kingdom
• Cooperation Partners: Professor Dr. Ivica Nakic; Professorin Dr. Angkana Rüland