Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials
Abstract
We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level N on Td. For every d,N≥2 and ε∈(0,1), we construct an explicit norming set with at most (C1(1+ε−1))d−1N1+ε points and norming constant at most (C2(1+ε−1))d−1, where C1,C2 are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent 1+ε: a sample of size at least C(d,ε)N1+εlog(2/η) norms the entire space simultaneously with probability at least 1−η, with a norming constant bounded by exp(Cdε−d) and independent of N. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant B>1 for the trigonometric polynomials with frequencies in {−1,0,1}d has at least (πd/(2elog B))d/2 points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in d when the norming constant is fixed.
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BibTeX
@misc{dai2026sampling,
title = {{Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials}},
author = {Feng Dai and Andriy Prymak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07679},
eprint = {2610.07679},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07679},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sampling discretization of the uniform norm for hyperbolic-cross trigonometric polynomials
AU - Feng Dai
AU - Andriy Prymak
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.07679
UR - https://arxiv.org/abs/2610.07679
AB - We study sampling discretization of the uniform norm for trigonometric polynomials with frequencies in a hyperbolic cross of level N on T^d. For every d,N≥2 and ε∈(0,1), we construct an explicit norming set with at most (C_1(1+ε^−1))^d−1N^1+ε points and norming constant at most (C_2(1+ε^−1))^d−1, where C_1,C_2 are absolute constants. We also prove that independent Haar-distributed points achieve the same exponent 1+ε: a sample of size at least C(d,ε)N^1+εlog(2/η) norms the entire space simultaneously with probability at least 1−η, with a norming constant bounded by exp(C_dε^−d) and independent of N. The deterministic construction combines uniformly stable de la Vallée Poussin sampling operators with a Smolyak-type combination identity. The random result follows from an abstract norming theorem for sums of spaces associated with commuting regular partitions, together with a multiscale approximation of hyperbolic-cross polynomials. Finally, we show that every norming set of constant B>1 for the trigonometric polynomials with frequencies in {−1,0,1}^d has at least (πd/(2elog B))^d/2 points. This rules out quadratic bounds in the dimension of the polynomial space with a prefactor growing only polynomially in d when the norming constant is fixed.
ER -