Bessel sequences of exponentials on fractal measures

Dutkay, Dorin Ervin; Han, Deguang; Weber, Eric

doi:10.1016/j.jfa.2011.06.018

Cited by 16 publications

(4 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is also worth mentioning that a widely open problem is to determine if μ 3 admits a Fourier frame or Riesz basis. Beurling dimension has been an indicator to see if such frame is possible to exist [6]. It was recently found that it is possible to construct an exponential Riesz sequence (note that a complete Riesz sequence will be a Riesz basis) with maximal Beurling dimension log 2/ log 3 [4].…”

Section: Definitions and Main Resultsmentioning

confidence: 99%

Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Lai

2023

Anal Math

View full text Add to dashboard Cite

Given an expansive matrix R ∈ M d (ℤ) and a finite set of digit B taken from ℤ d / R (ℤ d ). It was shown previously that if we can find an L such that ( R, B, L ) forms a Hadamard triple, then the associated fractal self-affine measure generated by ( R, B ) admits an exponential orthonormal basis of certain frequency set Λ, and hence it is termed as a spectral measure. In this paper, we show that if # B < |det( R )|, not only it is spectral, we can also construct arbitrarily sparse spectrum Λ in the sense that its Beurling dimension is zero.

show abstract

Section: Definitions and Main Resultsmentioning

confidence: 99%

Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Lai

2023

Anal Math

View full text Add to dashboard Cite

show abstract

“…In the end of the introduction, we remark that singular measures with Fourier frames but not an exponential basis were studied in [7,8,9,23,18]. In particular, in [18], such measures were shown to exist in abundance by relaxing the Hadamard triple condition proposed in [15] and [30].…”

Section: Introductionmentioning

confidence: 99%

Translational absolute continuity and Fourier frames on a sum of singular measures

Lai

2018

Journal of Functional Analysis

View full text Add to dashboard Cite

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+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 R θ with θ = ±π/2, the two-dimensional measure ρ θ (·) := (µ4 ×δ0)(·)+(δ0 ×µ16)(R −1 θ ·), supported on the union of x-axis and y = (cot θ)x, always admit a Fourier frame. Furthermore, we can find {e 2πi 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.

show abstract

“…Furthermore, some fractal measures are known not to admit any Fourier frames if the measures are non-uniform on the support [DL1]. Intensive researches on this question [DHSW,DHW1,DHW2,DL1,DL2,HLL] has been going on and one major advance was obtained recently in [DL2]. Dutkay and Lai introduced the almost-Parseval-frame condition for the self-similar measure and proved that if such condition is satisfied, the self-similar measure admits a Fourier frame.…”

Section: Introductionmentioning

confidence: 99%

Non-spectral fractal measures with Fourier frames

Lai¹,

Wang²

2017

J. Fractal Geom.

View full text Add to dashboard Cite

We generalize the compatible tower condition given by Strichartz to the almost-Parseval-frame tower and show that non-trivial examples of almost-Parseval-frame tower exist. By doing so, we demonstrate the first singular fractal measure which has only finitely many mutually orthogonal exponentials (and hence it does not admit any exponential orthonormal bases), but it still admits Fourier frames.2010 Mathematics Subject Classification. Primary 28A25, 42A85, 42B05.

show abstract

Bessel sequences of exponentials on fractal measures

Cited by 16 publications

References 20 publications

Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Arbitrarily Sparse Spectra for Self-Affine Spectral Measures

Translational absolute continuity and Fourier frames on a sum of singular measures

Non-spectral fractal measures with Fourier frames

Contact Info

Product

Resources

About