2023/10/02 by Riya Ghosh, Ghosh, Riya, A. Antony Selvan +1
Mathematics · #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2310.01141
For a window g∈ L2(ℝ), the subset of all lattice parameters (a, b)∈ ℝ2+ such that G(g,a,b)=\e2πib m⋅g(⋅-a k) : k, m∈ℤ\ forms a frame for L2(ℝ) is known as the frame set of g. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in \citecounter conjectured that if a0=\dfrac12m+1,~ b0=\dfrac2k+12,~k,m∈ ℕ,~kgt;m,~a0b0lt;1,then the Gabor system G(Q2, a, b) of the second order B-spline Q2 is not a frame along the hyperbolas ab=\dfrac2k+12(2m+1), for b∈ [b0-a0\dfrack-m2, b0+a0\dfrack-m2],for every a0, b0. Nielsen in \cite Nielsenthesis also conjectured that G(Q2, a,b) is not a frame for a=\dfrac12m,~b=\dfrac2k+12,~k,m∈ ℕ,~kgt;m,~ablt;1 with gcd(4m,2k+1)=1. In this paper, we prove that both conjectures are true.