Diagonal recovery from noisy Fourier--Hermite coefficients: exact orders of the error

@misc{solodky2026diagonal,
  title = {{Diagonal recovery from noisy Fourier--Hermite coefficients: exact orders of the error}},
  author = {S. G. Solodky and Y. A. Volynets},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15610},
  eprint = {2610.15610},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15610},
  note = {CC0},
}

TY  - UNPB
TI  - Diagonal recovery from noisy Fourier--Hermite coefficients: exact orders of the error
AU  - S. G. Solodky
AU  - Y. A. Volynets
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15610
UR  - https://arxiv.org/abs/2610.15610
AB  - We consider the recovery of functions of the weighted Wiener classes W_s^μ from perturbed Fourier--Hermite coefficients in the metric of L_∞(ℝ). For the summation methods of the class 𝒮^θ we find error estimates on W_s^μ. The estimates take into account the behaviour of the Hermite functions both in the oscillatory region and near the turning point: the maxima of functions with different indices are attained at different points, and the supremum of a sum over a block of indices is substantially smaller than the sum of the suprema. Owing to this, in particular, we establish a common lower bound for all finite diagonal methods; for s≠4/3 and p≠4/3 it matches the upper bound and gives the exact order of the error in this class, while at the critical points the gap between the bounds does not exceed (log(e/δ))^1/4 up to a constant. If, in addition, s>4/3 or p<4/3, then this order is also optimal among all recovery operators that use the whole sequence of data.
ER  -