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In this work, we analyze Gabor frames for the Weyl--Heisenberg group andwavelet frames for the extended affine group. Firstly, we give necessary andsufficient conditions for the existence of nonstationary frames of translates.Using these conditions, we give the existence of Gabor frames from theWeyl--Heisenberg group and wavelet frames for the extended affine group. Wepresent a representation of functions in the closure of the linear span of aGabor frame sequence in terms of the Fourier transform of window functions. Weshow that the canonical dual of frames of translates has the same structure. Anapproximation of inverse of the frame operator of nonstationary frames oftranslates is presented. It is shown that a nonstationary frame of translatesis a Riesz basis if it is linearly independent and satisfies approximation ofthe inverse frame operator. Finally, we give equivalent conditions for anonstationary sequence of translates to be linearly independent.

Nonstationary frames of translates and frames for the Weyl--Heisenberg group and the extended affine group