Stability of the shifts of global supported distributions

Abstract

For a tempered distribution with ℓ1 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 ℓ1 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 ℓ1 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. © 2001 Academic Press.