Low-dimensional approximation of uniformly bounded orthonormal systems
Abstract
We prove that if f1,…,fN is an uniformly bounded orthonormal system, then for all p∈[1,2) there is an estimate for its Kolmogorov widths: dn({f1,…,fN},Lp) ≳ min{1,n−1/pN1/2}, n≤ N/2. In the regime nasymp N this provides the lower bound N−αp with a sharp exponent αp:=1/p−1/2. Besides that, it follows that a good approximation of such ONS requires the dimension n≳ Np/2. Our second result is an approximation theorem for Fourier matrices of finite abelian groups (this includes the usual DFT matrices). For every η>0 there is a=a(η)>0 such that any Fourier matrix F of sufficiently large order has an approximation of rank N1−a with the row-wise ℓ1-error at most N1/2+η; the exponent 1/2 is sharp. Consequences include the approximation of the same kind for circulant matrices; the approximation of the trigonometric functions exp(2πi⟨ λ,x⟩), λ∈ Λ=K∩ℤd, K is symmetric convex, with dimension |Λ|1−a and an optimal error |Λ|−αp+η; bounds for widths of weighted Wiener classes.
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BibTeX
@misc{malykhin2026lowdimensional,
title = {{Low-dimensional approximation of uniformly bounded orthonormal systems}},
author = {Yuri Malykhin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37790},
eprint = {2609.37790},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37790},
}
RIS
TY - UNPB
TI - Low-dimensional approximation of uniformly bounded orthonormal systems
AU - Yuri Malykhin
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37790
UR - https://arxiv.org/abs/2609.37790
AB - We prove that if f_1,…,f_N is an uniformly bounded orthonormal system, then for all p∈[1,2) there is an estimate for its Kolmogorov widths: d_n({f_1,…,f_N},L_p) ≳ min{1,n^−1/pN^1/2}, n≤ N/2. In the regime nasymp N this provides the lower bound N^−α_p with a sharp exponent α_p:=1/p−1/2. Besides that, it follows that a good approximation of such ONS requires the dimension n≳ N^p/2. Our second result is an approximation theorem for Fourier matrices of finite abelian groups (this includes the usual DFT matrices). For every η>0 there is a=a(η)>0 such that any Fourier matrix F of sufficiently large order has an approximation of rank N^1−a with the row-wise ℓ_1-error at most N^1/2+η; the exponent 1/2 is sharp. Consequences include the approximation of the same kind for circulant matrices; the approximation of the trigonometric functions exp(2πi⟨ λ,x⟩), λ∈ Λ=K∩ℤ^d, K is symmetric convex, with dimension |Λ|^1−a and an optimal error |Λ|^−α_p+η; bounds for widths of weighted Wiener classes.
ER -