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^2gesum_rhofrac{|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|^2le 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.