Uniform Coercivity for the Báez–Duarte Frame Operator

This paper proves uniform coercivity of the Báez–Duarte frame operator on the closed span generated by the Báez–Duarte functions, within the Hilbert-space formulation of the Nyman–Beurling program. No spectral information about the Riemann zeta function is assumed or used. Building on the coercive quadratic-form mechanism introduced in A Coercive Quadratic Form from the Riemann–Weil Explicit Formula (Zenodo DOI: 10.5281/zenodo.18529291), we show that coercivity persists uniformly across scales when the problem is lifted to the Báez–Duarte frame setting. The Gram operator is decomposed into dyadic diagonal blocks and exponentially decaying off-diagonal blocks. Each diagonal block is approximated by a Toeplitz operator whose symbol is given by the autocorrelation of the sawtooth kernel. Uniform positivity of these Toeplitz blocks is established, while the remainder admits a summable decay bound yielding operator-norm control via a weighted Schur argument. As a consequence, the full frame operator is uniformly coercive: there exists a constant c>0c>0c>0 such that ⟨Sf,f⟩≥c∥f∥2\langle Sf,f\rangle \ge c\|f\|^2⟨Sf,f⟩≥c∥f∥2 for all vectors in the closed span, with no degradation in the infinite-dimensional limit. This uniform coercivity result provides the key functional-analytic input for the rigidity and canonical-system arguments developed in Energy Rigidity and Canonical Forcing of the Riemann Zeta Zeros (Zenodo DOI: 10.5281/zenodo.18529295) and synthesized in Coercivity, Canonical Systems, and Spectral Rigidity (Zenodo DOI: 10.5281/zenodo.18529297).