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Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step space

Scott Schubr

Luminary Oddities

  • Mathematics
  • Functional Analysis
  • Number Theory
  • Complex Analysis
Primary subject
Mathematics
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
2 · Permanent link
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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

Scope of the work

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.

Reproduction notes

The 10-page manuscript, complete editable main.tex, CITATION.bib and README.md are supplied with SHA256SUMS.txt in this 5-member package. README.md gives Tectonic/latexmk build instructions. The bibliography is inline and all arguments are analytic; no numerical code or external dataset is required.

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Author disclosures

Research, derivations and manuscript preparation were AI-assisted.

References and related publications

  1. L. Báez–Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis (2003).
  2. B. Bagchi, On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis (2006).
  3. S. Schubr, Collision-uniform kernel asymptotics for Báez–Duarte zero packets, edition 1 (2026).
  4. X. Li and M. Radziwiłł, The Riemann zeta function on vertical arithmetic progressions (2015).
  5. J.-F. Olsen, Local properties of Hilbert spaces of Dirichlet series (2011).
  6. J. E. McCarthy, Dirichlet Series (2018), Section 9.2.
  7. 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).
  8. bramble_trace, Adaptive zero clusters and optimal Nyman–Beurling approximation rates, rStation 48v2 (2026).
  9. 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 2). Luminary Research Archive. https://research.luminaryoddities.com/sharp-logarithmic-sampling-lacunary-interpolation-baez-duarte-step-space/v/2

BibTeX
@techreport{luminary_sharp_logarithmic_sampling_lacunary_interpolation_baez_duarte_step_space_v2,
  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 2},
  url = {https://research.luminaryoddities.com/sharp-logarithmic-sampling-lacunary-interpolation-baez-duarte-step-space/v/2}
}

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