2017/07/05 by Xiaoye Fu, Fu, Xiaoye, Chun‐Kit Lai +1 · 2 citations
Mathematics · #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1707.01545
openalex publication_date 2017/07/05 · openalex created_date 2017/07/14 · openalex updated_date 2026/07/28
A finite Borel measure μ in \mathbb Rd is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for L2(μ). It has been conjectured that a frame-spectral measure must be translationally absolutely continuous, which is a criterion describing the local uniformity of a measure on its support. In this paper, we show that if any measures ν and λ without atoms whose supports form a packing pair, then ν∗ λ+δt∗ν is translationally singular and it does not admit any Fourier frame. In particular, we show that the sum of one-fourth and one-sixteenth Cantor measure μ4+μ16 does not admit any Fourier frame. We also interpolate the mixed-type frame-spectral measures studied by Lev and the measure we studied. In doing so, we demonstrate a discontinuity behavior: For any anticlockwise rotation mapping Rθ with θ≠ ±π/2, the two-dimensional measure ρθ (⋅): = (μ4×δ0)(⋅)+(δ0×μ16)(Rθ-1⋅), supported on the union of x-axis and y=(\cot θ)x, always admit a Fourier frame. Furthermore, we can find \e2πi ⟨λ,x⟩\λ∈Λθ such that it forms a Fourier frame for ρθ with frame bounds independent of θ. Nonetheless, ρ±π/2 does not admit any Fourier frame.