A Short Note on the Frame Set of Odd Functions

Abstract

In this work we derive a simple argument which shows that Gabor systems consisting of odd functions of $d$ variables and symplectic lattices of density $2^d$ cannot constitute a Gabor frame. In the 1--dimensional, separable case, this is a special case of a result proved by Lyubarskii and Nes, however, we use a different approach in this work exploiting the algebraic relation between the ambiguity function and the Wigner distribution as well as their relation given by the (symplectic) Fourier transform. Also, we do not need the assumption that the lattice is separable and, hence, new restrictions are added to the full frame set of odd functions.

accepted: Bulletin of the Australian Mathematical Society; 12 pages; Version 3 makes use of symmetric time-frequency shifts. In this case the appearing phase factors are easier to handle. Also, the main result is extended to higher dimensions. [In Version 2 a mistake in the assumptions was corrected. The windows should be chosen from Feichtinger's algebra rather than from the Hilbert space L2.]

References in corpus (2)

• Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions
• Minimal Operator Norms via Minimal Theta Functions

Cited by in corpus (6)

• Maximal Theta Functions -- Universal Optimality of the Hexagonal Lattice for Madelung-Like Lattice Energies
• On the frame property of Hermite functions and exploration of their frame sets
• Some curious results related to a conjecture of Strohmer and Beaver
• A variational principle for Gaussian lattice sums
• On a fundamental barrier of the Wirtinger criterion for Gabor systems with odd functions
• The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators