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

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.