Polynomial cores and inertia under Carleman moment bounds

Overview

Abstract

We study ordinary polynomial approximation in a joint norm containing a holomorphic product derivative on a full horizontal strip and a real-axis observation. A Carleman condition on the combined horizontal moments gives a polynomial core for finite-order entire divisors, with every interior zero multiplicity retained and with no boundary-zero separation. A finite moment version gives an explicit terminal-clock estimate for the original complex-coordinate graph. For factorial--logarithmic moment budgets we prove a universal core guarantee through exponent one and give finite-order, real-zero counterexamples beyond it, using their own physical observation weights. For a separate real signed-measure quadratic form, a negative interval gives whole negative polynomial spaces of order (log N)^(1-α) for 0≤α<1, and loglog N at α=1, in each parity. Slowly varying Jacobi models give matching orders in this finite-measure class. Finally, a double-exponential spatial kernel has the required square-root-logarithmic Fourier moments for its actual physical weight. Its joint core at every width coexists with growing negative physical polynomial inertia at later widths, despite its positive all-depth root hierarchy. Classical quasianalyticity, Dirichlet cyclicity, scalar approximation and orthogonal-polynomial results are explicitly credited.

Keywords Carleman moment bounds · polynomial approximation · holomorphic strip · Dirichlet space · signed measures · negative inertia · Jacobi matrices