Signed frames and Hadamard products of Gram matrices Original Research Article

This paper concerns (redundant) representations in a Hilbert space H of the form image f=∑jcjleft angle bracketf,φjright-pointing angle bracketφj for allfset membership, variantH.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 image image 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.