Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants
Abstract
Let dk>0 and let Qk be a polynomial of degree rk with nonnegative Chebyshev coefficients. We study locally uniform limits of fk(z)=CkQk(cos(dkz)), where Ck>0. If fk→ F locally uniformly and F(0)>0, the associated positive lattice measures converge weakly, their second moments converge, and their quadratic tails are uniformly integrable. If additionally all zeros of Qk lie in [−1,1), dk→0, and F has order below two, then for every A>0 with F(A)F(−A)≠0, limsupk→∞ rkdk2≤ 4∑γn>Aγn−2, where {±γn} are the nonzero real zeros of F, counted with multiplicity. If, moreover, fk→ F locally uniformly with F(0)>0, all zeros of Qk lie in [−1,1), and F has no nonzero real period, has order below two, and is not of finite exponential type, then dk→0, rkdk→∞, and rkdk2→0. Equivalently, dk−1=o(rk) and rk=o(dk−2). Boundary examples show the role of the hypotheses and the Gaussian boundary at order two.
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BibTeX
@misc{stodolsky2026meshdegree,
title = {{Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants}},
author = {Vasily Stodolsky},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28792},
eprint = {2608.28792},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28792},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Mesh-Degree Rigidity for Positive Chebyshev-Fourier Approximants
AU - Vasily Stodolsky
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28792
UR - https://arxiv.org/abs/2608.28792
AB - Let d_k>0 and let Q_k be a polynomial of degree r_k with nonnegative Chebyshev coefficients. We study locally uniform limits of f_k(z)=C_kQ_k(cos(d_kz)), where C_k>0. If f_k→ F locally uniformly and F(0)>0, the associated positive lattice measures converge weakly, their second moments converge, and their quadratic tails are uniformly integrable. If additionally all zeros of Q_k lie in [−1,1), d_k→0, and F has order below two, then for every A>0 with F(A)F(−A)≠0, limsup_k→∞ r_kd_k^2≤ 4∑_γ_n>Aγ_n^−2, where {±γ_n} are the nonzero real zeros of F, counted with multiplicity. If, moreover, f_k→ F locally uniformly with F(0)>0, all zeros of Q_k lie in [−1,1), and F has no nonzero real period, has order below two, and is not of finite exponential type, then d_k→0, r_kd_k→∞, and r_kd_k^2→0. Equivalently, d_k^−1=o(r_k) and r_k=o(d_k^−2). Boundary examples show the role of the hypotheses and the Gaussian boundary at order two.
ER -