Bracket Products for Weyl-heisenberg Frames
نویسنده
چکیده
Abstract. We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product specifically geared to Weyl-Heisenberg frames. This bracket product has all the properties of a standard inner product including Bessel’s inequality, a Riesz Representation Theorem, and a Gram-Schmidt process which turns a sequence of functions (gn) into a sequence (en) with the property that (Emben)m,n∈Z is orthonormal in L (R). Armed with this inner product, we obtain several results concerning WeylHeisenberg frames. First we see that fiberization in this setting takes on a particularly simple form and we use it to obtain a compressed representation of the frame operator. Next, we write down explicitly all those functions g ∈ L(R) and ab = 1 so that the family (EmbTnag) is complete in L(R). One consequence of this is that for functions g supported on a half-line [α,∞) (in particular, for compactly supported g), (g, 1, 1) is complete if and only if sup0≤t<a|g(t − n)| 6 = 0 a.e. Finally, we give a direct proof of a result hidden in the literature by proving: For any g ∈ L(R), A ≤ ∑ n |g(t − na)|2 ≤ B is equivalent to (Em/ag) being a Riesz basic sequence.
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