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We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds forfinite Gabor systems generated by square-integrable functions with certainbehavior at infinity. These functions include functions ultimately decayingfaster than any exponential function, as well as square-integrable functionsultimately analytic and whose germs are in a Hardy field. Two classes of thelatter type of functions are the set of square-integrablelogarithmico-exponential functions and the set of square-integrable Pfaffianfunctions. We also prove the HRT conjecture for certain finite Gabor systemsgenerated by positive functions.

Linear Independence of Finite Gabor Systems Determined by Behavior at Infinity