English

Quantitative bounds for unconditional pairs of frames

Functional Analysis 2022-12-05 v1 Numerical Analysis Numerical Analysis

Abstract

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [SB2]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that there is a universal constant κ>0\kappa>0 so that for all C,β>0C,\beta>0 and NNN\in\mathbb{N} the following is true. Let (xj)j=1N(x_j)_{j=1}^N and (fj)j=1N(f_j)_{j=1}^N be sequences in a finite dimensional Hilbert space which satisfy xj=fj\|x_j\|=\|f_j\| for all 1jN1\leq j\leq N and j=1Nεjx,fjxjCx, for all x2M and εj=1.\Big\|\sum_{j=1}^N \varepsilon_j\langle x,f_j\rangle x_j\Big\|\leq C\|x\|, \qquad\textrm{ for all $x\in \ell_2^M$ and $|\varepsilon_j|=1$}. If the frame operator for (fj)j=1N(f_j)_{j=1}^N has eigenvalues λ1...λM\lambda_1\geq...\geq\lambda_M and λ1βM1j=1Mλj\lambda_1\leq \beta M^{-1}\sum_{j=1}^M\lambda_j then (fj)j=1N(f_j)_{j=1}^N has Bessel bound κβ2C\kappa \beta^2 C. The same holds for (xj)j=1N(x_j)_{j=1}^N.

Keywords

Cite

@article{arxiv.2212.00947,
  title  = {Quantitative bounds for unconditional pairs of frames},
  author = {Peter Balazs and Daniel Freeman and Roxana Popescu and Michael Speckbacher},
  journal= {arXiv preprint arXiv:2212.00947},
  year   = {2022}
}

Comments

19 pages

R2 v1 2026-06-28T07:20:05.695Z