Research Article · Preprint Mathematics
Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step space
- Primary subject
- Mathematics
- Subject subfields
- Functional Analysis · Number Theory · Complex Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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.
Keywords Báez–Duarte approximation · harmonic-step Mellin space · logarithmic sampling threshold · Newton–Hermite packets · lacunary interpolation · Dirichlet series
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Research, derivations and manuscript preparation were AI-assisted.
References and related publications
- 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).
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step space (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/sharp-logarithmic-sampling-lacunary-interpolation-baez-duarte-step-space/v/1
BibTeX
@techreport{luminary_sharp_logarithmic_sampling_lacunary_interpolation_baez_duarte_step_space_v1,
title = {Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step space},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/sharp-logarithmic-sampling-lacunary-interpolation-baez-duarte-step-space/v/1}
}Loading discussion…