发表机构
SDU University; Institute of Mathematics and Mathematical Modeling（SDU大学; 数学与数学建模研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Laguerre变换的加权系数估计，建立了由Laguerre平移生成的光滑模与系数范数之间的双边界，并在$p=2$时得到等价性，证明了乘子权重的最优性，恢复了Jackson和Bernstein型估计及Lipschitz类的$L^2$刻画。
AI 中文摘要
我们研究了由Laguerre平移生成的光滑模所刻画的Laguerre变换的加权系数估计。对于每个整数$r\ge1$和$1\le p\le2$，加权$\ell^{p'}$范数$\min\{1,nt\}^{r}\widehat f_\alpha(n)$被$r$阶光滑模所界定；对于$2\le p\le\infty$，在相同的加权序列尺度上反向不等式成立。在$p=2$时，这些估计产生双边等价，并且测试单个Laguerre多项式证明了乘子权重的最优性（直至常数）。证明结合了Hausdorff--Young不等式与平移乘子的平均下界。作为推论，我们恢复了Jackson型和Bernstein型估计，以及对于$0<\beta<r$的Lipschitz类的已知$L^2$逼近刻画。我们还给出了对于一般$p$的相应单边系数尾准则，以及饱和和端点陈述。
英文摘要
We study weighted coefficient estimates for the Laguerre transform in terms of moduli of smoothness generated by the Laguerre translation. For every integer $r\ge1$ and $1\le p\le2$, a weighted $\ell^{p'}$ norm of $\min\{1,nt\}^{r}\widehat f_α(n)$ is bounded by the modulus of order $r$; for $2\le p\le\infty$ 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.
Comments31 pages
