Signed frames and Hadamard products of Gram Matrices

Abstract

This paper concerns (redundant) representations in a Hilbert space $H$ of the form $$ f = sum_j c_jinpro{f,phi_j}phi_j, qquad forall fin H. $$ 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 $c_j$ are unique and the geometric interpretation of negative $c_j$. 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 $v_1,ldots,v_ninCC^d$ $$ rank([inpro{v_i,v_j}^roverline{inpro{v_i,v_j}}^s]) = min{{r+d-1choose d-1}{s+d-1choose d-1},n}, qquad r,sge0. $$ 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 frames in a finite dimensional space.