Weighted tight frames of exponentials on a finite interval

Abstract

Given a finite intervalI⊂R, a characterization is given for those discrete sets of real numbers Λ and associated sequences {c _λ}_λ∈Λ, withc _λ>0, having the properties that every functionf∈L ²(I) can be expanded inL ²(I) as the unconditionally convergent series

$$f = \sum\limits_{\lambda \in \Lambda } {\hat f} (\lambda )c_\lambda e^{2\pi i\lambda x} $$

and that the range of the mappingL ²(I)→L ²_μ :f→f has finite codimension inL ²_μ , iff denotes the Fourier transform off and μ is the measure μ = ∑_λ∈Λ c _λ δ_λ.