Binary Frames

We investigate frame theory over the binary field Z2, following work of Bodmann, Le, Reza, Tobi and Tomforde. We consider general finite dimensional vector spaces V over Z2 equipped with an (indefinite) inner product (·, ·)V which can be an arbitrary bilinear functional. We characterize precisely when two such spaces (V, (·, ·)V ) and (W, (·, ·)W ) are unitarily equivalent in the sense that there is a linear isomorphism between them that preserves inner products. We do this in terms of a computable invariant we term the matricial spectrum of such a space. We show that an (indefinite) inner product space (V, (·, ·)V ) is always unitarily equivalent to a subspace of (Z2 ), 〈·, ·〉) for sufficiently large n, where 〈·, ·〉 denotes the standard dot product bilinear functional on Z2 . This embedding theorem reduces the general theory to the theory of subspaces of (Z2 , 〈·, ·〉. We investigate the existence of dual frames and Parseval frames for vector spaces over Z2. We characterize precisely when a general (indefinite) inner product space (V, (·, ·)V ) satisfies a version of the Riesz Representation Theorem. We also consider the subspaces On consisting of all vectors x in (Z2 , 〈·, ·〉) with 〈x, x〉 = 0, and we show that On has a Parseval frame for odd n, and for even n it has a subspace of codimension one that has a Parseval frame. Introduction Frames on inner product spaces (Hilbert spaces) over the fields of real or complex numbers have been well studied and are a useful tool in both theoretical and applied mathematics. Formally, a frame F for an inner product space H over R or C is a sequence of vectors {xn} in H indexed by a countable index set J for which there exist constants 0 < A ≤ B < ∞ such that, for every x ∈ H, A‖x‖2 ≤ ∑ n∈J | 〈 x, xn 〉 |2 ≤ B‖x‖2. A frame over R or C is called tight if A = B and is called a Parseval frame if A = B = 1. If H is finite dimensional then finite frames over R or C are easily characterized as finite indexed sets whose linear span is H. Every frame {xn} in H over R or C has a dual frame {yn} (generally non-unique) which has the reconstruction property that for all x in H we have x = ∑ n∈J 〈 x, yn 〉xn. Parseval frames are self-dual and are characterized by the property that x = ∑ n∈J 〈 x, xn 〉xn , x ∈ H. We refer to [2], [4], [5] for background and exposition of the R and C frame theory. The purpose of this article is to build onto the previous work by Bodmann, Le, Reza, Tobi and Tomforde [1] in which they introduce frame theory over the finite field of characteristic 2. They considered Z2 equipped with the standard dot product, which we will denote by 〈·, ·〉, as an indefinite inner product and proved that (Z2 , 〈·, ·〉) has a number of properties such as the Riesz Representation Theorem. Some aspects of the theory remain nearly the same as in the inner product space theory over R, but others are strikingly different. The formal definition of the dot product on Z2 for x = (x1, x2, . . . , xn) and y = (y1, y2, . . . , yn) is 〈x, y〉 := ∑ xiyi. This is an indefinite inner product since many vectors x, such as (1, 1), have the property that 〈x, x〉 = 0, which is a major reason for the difference with the R theory. There are many characterizations of frames on Rn, and any of them could in principle be taken as a starting definition for developing the theory. But for Z2 the only definition that makes sense is that a frame is simply a spanning set for the space, and this is the approach taken in [1]. This leads to special types of frames, such as bases, Parseval frames and duals 2000 Mathematics Subject Classification. 46L99, 42C15, 47A05, 47L35, 47L05.