1. Quantitative Uniqueness Properties for L2 Functions with Fast Decaying, or Sparsely Supported, Fourier Transform
- Author
-
Benjamin Jaye and Mishko Mitkovski
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Extension (predicate logic) ,Characterization (mathematics) ,01 natural sciences ,Set (abstract data type) ,symbols.namesake ,Fractal ,Fourier transform ,Uniqueness theorem for Poisson's equation ,0103 physical sciences ,Key (cryptography) ,symbols ,010307 mathematical physics ,Uniqueness ,0101 mathematics ,Mathematics - Abstract
This paper builds upon two key principles behind the Bourgain–Dyatlov quantitative uniqueness theorem for functions with Fourier transform supported in an Ahlfors regular set. We first provide a characterization of when a quantitative uniqueness theorem holds for functions with very quickly decaying Fourier transform, thereby providing an extension of the classical Paneah–Logvinenko–Sereda theorem. Secondly, we derive a transference result which converts a quantitative uniqueness theorem for functions with fast decaying Fourier transform to one for functions with Fourier transform supported on a fractal set. In addition to recovering the result of Bourgain–Dyatlov, we obtain analogous uniqueness results for denser fractals.
- Published
- 2021