AI 中文总结
研究最大乘积广义采样算子在 \(L^{p}\) 范数下的收敛性与保形性质，通过\(\tau\)-模建立逼近误差估计，证明其 \(L^{p}\) 收敛，还将特定核的保形结果推广到更广泛核类，部分保持函数单调性。
AI 中文摘要
本文研究最大乘积广义采样算子在 \(L^{p}\) 范数下的收敛性及某些保形性质。对于定义在 \([-1,1]\) 上的非负有界函数，在 \(1 \leq p < +\infty\) 时，借助\(\tau\)-模建立了 \(L^{p}\) 范数下逼近误差的定量估计。证明了最大乘积广义采样算子 \(L^{p}\) 收敛到 \([-1,1]\) 上可测、有界且Riemann可积的非负函数。最后将Coroianu和Gal对特定核的保形结果推广到更广泛的光滑中心钟形核类。在合适假设下，证明了该算子部分保持 \([0,1]\) 上非减或非增函数 \(f:[0,1] \to \R_{}\) 的单调性。
英文摘要
In this paper, we investigate the convergence in the $L^{p}$-norm and certain shape-preserving properties of the max-product generalized sampling operators. More precisely, we establish quantitative estimates for the approximation error in the $L^{p}$-norm, for $ 1 \le p < +\infty$, in the case of non-negative and bounded functions defined on $[-1,1]$. These estimates are derived by means of the so-called $τ$-modulus, an averaged modulus of smoothness introduced by Sendov and Popov. As a direct consequence, we prove that the max-product generalized sampling operators $L^{p}$-converge to non-negative functions that are measurable, bounded and Riemann integrable on the interval $[-1,1]$. In the final section, we extend several shape-preserving results of Coroianu and Gal, originally established for specific kernels (such as the sinc/Whittaker and Fejér kernels), to the broader class of smooth centered bell-shaped kernels. Under suitable assumptions on the kernel, we prove that the max-product generalized sampling operators partially preserve the monotonicity of any function $f:[0,1] \rightarrow \R_{0}^{+}$ that is either non-decreasing or non-increasing on $[0,1]$.
