Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step space
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Abstract
We prove a sharp distinction between discrete and continuous sampling in the harmonic-step Mellin space underlying Báez--Duarte approximation. At real part one, height-normalized derivatives of order at most R admit a uniform logarithmically weighted estimate on every separated sequence precisely for exponents p≥ 2R+2. The continuous-height estimate instead holds precisely for p>2R+1. An explicit family supported at powers of two proves the discrete threshold while retaining the exact adjacent-endpoint difference of the step basis. Inside the strip 1/2<Re s<1, we prove count-independent sampling for Newton--Hermite packets with bounded size and arbitrary internal collisions. Complete coefficient-tail estimates then yield a full infinite Riesz Gram bound, a bounded interpolation operator and an exact projection formula for sufficiently lacunary packets. The zero-packet application controls all cross-packet terms in the native approximation norm. Finally, arbitrarily sparse interpolating nodes can admit a common-zero vector arbitrarily close to the target while excluding every nonzero Dirichlet numerator absolutely convergent at their real part.
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Research details
- Subjects
- Mathematics · Number Theory · Complex Analysis · Functional Analysis
- Keywords
- Báez–Duarte approximation, harmonic-step Mellin space, logarithmic sampling threshold, Newton–Hermite packets, lacunary interpolation, Dirichlet series
- Manuscript date
- 2026-10-26
- Reader reuse
- No additional permission granted
Scope
In the harmonic-step Mellin norm, separated height sampling at real part one has exact logarithmic threshold p≥2R+2 and the continuous-height estimate has threshold p>2R+1. Bounded collision packets in the open strip admit count-independent sampling and sufficiently lacunary infinite interpolation with the stated arithmetic-regularity obstruction.
Disclosures
Research, derivations and manuscript preparation were AI-assisted.
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References and related resources
- L. Báez–Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis (2003).
- B. Bagchi, On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis (2006).
- S. Schubr, Collision-uniform kernel asymptotics for Báez–Duarte zero packets, edition 1 (2026).
- X. Li and M. Radziwiłł, The Riemann zeta function on vertical arithmetic progressions (2015).
- J.-F. Olsen, Local properties of Hilbert spaces of Dirichlet series (2011).
- J. E. McCarthy, Dirichlet Series (2018), Section 9.2.
- span_of_dilates, The Báez–Duarte subspace has codimension equal to the number of zeta zeros off the critical line, with distance and rate bounds not assuming RH, rStation 1v5 (2026).
- bramble_trace, Adaptive zero clusters and optimal Nyman–Beurling approximation rates, rStation 48v2 (2026).
- S. Ferri, J. Galindo and C. Gómez, Sampling almost periodic and related functions (2020).