Obstructions for Gabor frames of the second-order B-spline: Obstructions for Gabor frames of the second-order B-spline - Ghosh, Riya; Selvan, A. Antony (Advances in computational mathematics, 2025)
For a window g ∈ L 2 ( R ) , the subset of all lattice parameters ( a , b ) ∈ R + 2 such that G ( g , a , b ) = { e 2 π i b m · g ( · - a k ) : k , m ∈ Z } forms a frame for L 2 ( 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 (J. Fourier Anal. Appl. 22 , 1440–1451, 2016) conjectured that if a 0 = 1 2 m + 1 , b 0 = 2 k + 1 2 , k , m ∈ N , k > m , a 0 b 0 < 1 , then the Gabor system G ( Q 2 , a , b ) of the second-order B-spline Q 2 is not a frame along the hyperbolas a b = 2 k + 1 2 ( 2 m + 1 ) , for b ∈ b 0 - a 0 k - m 2 , b 0 + a 0 k - m 2 , for every a 0 , b 0 . Nielsen in (2015) also conjectured that G ( Q 2 , a , b ) is not a frame for a = 1 2 m , b = 2 k + 1 2 , k , m ∈ N , k > m , a b < 1 with gcd ( 4 m , 2 k + 1 ) = 1 . In this paper, we prove that both Conjectures are true.