Moduli of smoothness and growth of the Laguerre transform: two-sided estimates
Abstract
We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer r≥1 and 1≤ p≤2, a weighted ℓp' norm of min{1,nt}rf̂α(n) is bounded by the modulus of order r; for 2≤ p≤∞ the reverse direction holds on the same weighted sequence scale. At p=2 these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff--Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known L2 approximation characterisation of the Lipschitz classes for 0<β<r. We also give the corresponding one-sided coefficient-tail criteria for general p, together with saturation and endpoint statements.
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BibTeX
@misc{kakharman2026moduli,
title = {{Moduli of smoothness and growth of the Laguerre transform: two-sided estimates}},
author = {Nurbek Kakharman and Niyaz Tokmagambetov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08167},
eprint = {2610.08167},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08167},
}
RIS
TY - UNPB
TI - Moduli of smoothness and growth of the Laguerre transform: two-sided estimates
AU - Nurbek Kakharman
AU - Niyaz Tokmagambetov
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08167
UR - https://arxiv.org/abs/2610.08167
AB - We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer r≥1 and 1≤ p≤2, a weighted ℓ^p' norm of min{1,nt}^rf̂_α(n) is bounded by the modulus of order r; for 2≤ p≤∞ the reverse direction holds on the same weighted sequence scale. At p=2 these estimates yield a two-sided equivalence, and testing individual Laguerre polynomials proves optimality of the multiplier weight up to constants. The proof combines Hausdorff--Young inequalities with an averaged lower bound for the translation multiplier. As consequences we recover Jackson- and Bernstein-type estimates and the known L^2 approximation characterisation of the Lipschitz classes for 0<β<r. We also give the corresponding one-sided coefficient-tail criteria for general p, together with saturation and endpoint statements.
ER -