BINARY FRAMES Academic Article uri icon

abstract

  • © 2015 University of Houston. We investigate frame theory over the binary field Z2, following work of Bodmann, Le, Reza, Tobin 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 character-ize 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 (Zn2); h.;.i) for suffciently large n, where h.;.i denotes the standard dot prod-uct bilinear functional on Zn2. This embedding theorem reduces the general theory to the theory of subspaces of (Zn2; h.;.i). We investigate the existence of dual frames and Parseval frames for vector spaces over Z2. We charac-terize 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 (Zn2; h.;.i) with hx; xi = 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.

author list (cited authors)

  • Hotovy, R., Larson, D. R., & Scholze, S.

complete list of authors

  • Hotovy, Ryan||Larson, David R||Scholze, Sam

publication date

  • January 2015