Signed frames and Hadamard products of Gram matrices

Abstract

This paper concerns (redundant) representations in a Hilbert space H of the form f = (j)Sigma c(j) <f, Phi (j)> Phi (j) For Allf is an element of H . These are more general than those obtained from a tight frame, and we develop a general theory based on what are called signed frames. We are particularly interested in the cases where the scaling factors cj are unique and the geometric interpretation of negative cj. This is related to results about the invertibility of certain Hadamard products of Gram matrices which are of independent interest, e.g., we show for almost every nu(1).....,nu(n) is an element of C-d rank ([<nu(1) ,nu(j)> (r) <nu(i), nu(j)>(s)]) = min {((r+d-1)(d-1))((s+d-1)(d-1)).n}, r, s greater than or equal to 0. Applications include the construction of tight frames of bivariate Jacobi polynomials on a triangle which preserve symmetries, and numerical results and conjectures about the class of tight signed frames in a finite-dimensional space. (C) 2002 Elsevier Science Inc. All rights reserved.