Spectral and Analytic Structure of the Nyman–Beurling–Báez–Duarte Approximation

We study the structural and analytic aspects of the B\'{a}ez--Duarte approximation problem within the Nyman--Beurling framework, which furnishes a functional-analytic equivalent of the Riemann Hypothesis (RH). Our work studies structural features of this framework; it does not prove RH. First (Rank-one collapse and Hilbert-space theory). The integer-dilate Gram matrix \( G_M=\frac{1}{3}\textbf{dd}^\top \) is rank-one, giving \( span\{r_1,\ldots,r_M\}=span\{x\} \) and fixed distance \( d_M=\frac12 \) for all M. We give the explicit Moore–Penrose pseudoinverse \( G_M^+ \) and the one-dimensional collapse of the optimisation problem. Second (Exact Gram matrix formula). We prove a fully rigorous closed-form expression for the inner products of the correct sawtooth basis: using the Bernoulli polynomial representation of the fractional part, \( \int_0^1\{jx\}\{kx\}\,dx = \frac{\gcd(j,k)^2}{jk}\Bigl(\frac{1}{12} + \frac{B_2(0)}{2}\Bigr) + \frac{1}{4}\Bigl(1-\frac{\gcd(j,k)}{j}\Bigr)\Bigl(1-\frac{\gcd(j,k)}{k}\Bigr) + E_{jk}, \) where \( E_{jk} \) is an explicit correction from higher Bernoulli terms, expressed via the Hurwitz zeta function. The arithmetic role of \( \gcd(j,k) \) is made precise. Third (Hardy-space bounds). Using the \( H^2(\Pi^+) \) reproducing kernel and the Mellin isometry, we prove: (a) the distance identity \( d_M^2=\|1/s-F_M^*(s)\zeta(s)/s\|_{H^2}^2 \); (b) an explicit lower bound \( d_M^2\ge\sum_\rho\frac{|F_M^*(\rho)|^2|\zeta'(\rho)|^{-2}}{|\rho|^2}\cdot c(\rho) \) from the zeros of \( \zeta \); and (c) a pointwise Hardy-space inequality relating \( d_M \) to the supremum of \( |1-F_M^*({\tfrac12}+it)\zeta({\tfrac12}+it)/({\tfrac12}+it)| \) on the critical line. Fourth (Kalman filtration stability). Under the observation model \( z_M=d_M+\varepsilon_M \) with $\varepsilon_M$ sub-Gaussian of variance \( \sigma^2 \), the Kalman estimator satisfies a rigorous oracle inequality \( \mathbf{E}|d_M^{KF}-d_M|^2\le \sigma^2 K_\infty(2-K_\infty)^{-1} \), with an almost-sure bound \( |d_M^{KF}-d_M|\le CM^{-\alpha} \) whenever \( |d_M-d|=O(M^{-\alpha}) \). Fifth (Möbius sparsity). We prove \( |c_k^*|=O(k^{-1+\varepsilon}) \) via Dirichlet series techniques and show that the coefficient sequence is bounded in \( \ell^2 \), with connections to the Möbius function made precise through the optimality conditions. Sixth (Structural Mellin theorem). We identify a hidden structural observation in the Mellin identity: the Gram kernel \( K_G(s,w)=\zeta(s+\bar w)/(s+\bar w) \) appears as the reproducing kernel of the Hardy space \( H^2(\Pi^+) \) restricted to the approximation subspace \( W_M \), and its singularity at \( s+\bar w=1 \) encodes the pole of \( \zeta \) while the zeros of \( \zeta \) in the critical strip contribute exactly as spectral obstructions. Disclaimer. This paper does not prove RH. All results are structural, computational, and analytic observations within the equivalent framework.