# Signed frames and Hadamard products of Gram matrices

## by Shayne Waldron and Irine Peng

## 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

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