Stability of the Shifts of Global Supported Distributions

For a tempered distribution with l 1 decay, we characterize its stable shifts via its Fourier transform and via a shift-invariant space of summable sequences. Also we show that if the tempered distribution with l 1 decay has stable shifts, then we can recover all distributions in V , the space of all linear combinations of its shifts using bounded sequences, in a stable way using C dual functions with l 1 decay at infinity. If, additionally, that tempered distribution is compactly supported, then the above C dual functions can be chosen to have exponential decay at infinity.