Frames of Translates
نویسندگان
چکیده
Frames consisting of translates of a single function play an important role in sampling theory as well as in wavelet theory and Gabor analysis. We give a necessary and sufficient condition for a subfamily of regularly spaced translates of a function φ ∈ L(R), (τnbφ)n∈Λ, Λ ⊂ Z, to form a frame (resp. Riesz basis) for its closed linear span. One consequence is that if Λ ⊂ N , then this family is a frame if and only if it is a Riesz basis. For the case of arbitrary translates of a function φ ∈ L(R) we show that for sparse sets, having an upper frame bound is equivalent to the family being a frame sequence. Also we give some relatively mild density conditions will yield frame sequences. Finally, we us the fractional Hausdorff dimension to identify classes of exact frame sequences.
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