Time-frequency concentration of generating systems

Uncertainty principles for generating systems {en} ∞ n=1 ⊂ L (R) are proven and quantify the interplay between l(N) coefficient stability properties and time-frequency localization with respect to |t| power weight dispersions. As a sample result, it is proven that there does not exist a unit-norm Schauder basis nor a frame {en} ∞ n=1 for L (R) such that the two dispersion sequences ∆(en), ∆(ên) and one mean sequence μ(en) are bounded. On the other hand, it is constructively proven that there exists a unit-norm exact system {fn} ∞ n=1 in L(R) for which all four of the sequences ∆(fn), ∆(f̂n), μ(fn), μ(f̂n) are bounded.