# Signed frames and Hadamard products of Gram matrices

## Abstract:

This paper concerns (redundant) representations in a Hilbert space $H$ of the form $$f = \sum_j c_j\inpro{f,\phi_j}\phi_j, \qquad \forall f\in H.$$ 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 $v_1,\ldots,v_n\in\CC^d$ $$\rank([\inpro{v_i,v_j}^r\overline{\inpro{v_i,v_j}}^s]) = \min\{{r+d-1\choose d-1}{s+d-1\choose d-1},n\}, \qquad r,s\ge0.$$ 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.

Keywords: frames, wavelets, signed frames, Hadamard product, Gram matrix, generalised Hermitian forms, multivariate Jacobi polynomials, Lauricella functions

Math Review Classification: 05B20, 41A65, 42C15 (primary), 11E39, 33C50, 33C65, 42C40 (secondary)

Length: 22 pages

Last updated: 20 April 2001

Status: submitted