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• W2963224185 abstract "A finite Borel measure μ in R d is called a frame-spectral measure if it admits an exponential frame (or Fourier frame) for L 2 ( 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 + δ t ⁎ 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 μ 4 + μ 16 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 R θ with θ ≠ ± π / 2 , the two-dimensional measure ρ θ ( ⋅ ) : = ( μ 4 × δ 0 ) ( ⋅ ) + ( δ 0 × μ 16 ) ( R θ − 1 ⋅ ) , supported on the union of x -axis and y = ( cot ⁡ θ ) x , always admit a Fourier frame. Furthermore, we can find { e 2 π i 〈 λ , x 〉 } λ ∈ Λ θ such that it forms a Fourier frame for ρ θ with frame bounds independent of θ . Nonetheless, ρ ± π / 2 does not admit any Fourier frame." @default.
• W2963224185 created "2019-07-30" @default.
• W2963224185 creator A5030019449 @default.
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• W2963224185 date "2018-05-01" @default.
• W2963224185 modified "2023-10-18" @default.
• W2963224185 title "Translational absolute continuity and Fourier frames on a sum of singular measures" @default.
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• W2963224185 doi "https://doi.org/10.1016/j.jfa.2017.10.021" @default.
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