2024/07/06 by Yao-Qiang Li, Li, Yao-Qiang
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2407.05101
openalex publication_date 2024/07/06 · openalex created_date 2024/07/10 · openalex updated_date 2026/07/28
We study the spectrality of a class of infinite convolutions in ℝd, generalizing a result given by Li, Miao and Wang in 2022 from ℝ to ℝd. This allows us to easily construct spectral measures with and without compact supports in ℝd, and motivates us to systematically study the supports of infinite convolutions. In particular, we give a sufficient and necessary condition for infinite convolutions to exist with compact supports, generalizing a related well-known result which is widely used. After giving strong relations between supports of infinite convolutions and sets of infinite sums, we study the closedness and fractal dimensions of infinite sums of union sets in order to deal with non-compact supports of infinite convolutions. As an application of these new tools, we deduce that there are spectral measures with and without compact supports of arbitrary Hausdorff and packing dimensions in ℝd, generalizing another result given by Li, Miao and Wang in 2022 from ℝ to ℝd.