On Characterizations of Fourier Frames and Tilings

for all f ∈ H.A and B are called frame bounds.The sequence is called a tight frame if A = B. The sequence is called Bessel if the second inequality above holds. In this case, B is called the Bessel bound. Frames were first introduced by Duffin and Schaeffer [1] in the context of nonharmonic Fourier series, and today they have applications in a wide range of areas. A frame can be considered as a generalized basis in the sense that every element inH can bewritten as a linear combination of the frame elements. In this paper, we consider Fourier frames for a special separable Hilbert space. Let Ω ⊂ Rd have positive Lebesgue measure m(Ω) > 0 and let Λ be a discrete subset of R. The inner product and the norm on L2(Ω) are