AI 中文总结
该研究提出Bernoulli–Strang–Fix条件及其广义形式，用于分析采样Kantorovich算子的多项式再生、逼近与收敛性质，拓展其信号预测应用，并通过数值算例验证理论结果。
AI 中文摘要
本文引入了针对向量值生成元$\boldsymbol{\phi}=(\phi_0,\dots,\phi_{\rho-1})$与周期非均匀采样集$X$的Bernoulli–Strang–Fix条件及其广义形式。我们利用这些条件建立了与$(\boldsymbol{\phi},X)$关联的采样Kantorovich算子在指定阶数内的精确与渐近多项式再生性质，详细分析了这类算子的逼近特性与收敛行为。此外，我们展示了其在基于有限个过去局部平均样本的信号预测中的应用，证明采样Kantorovich算子也可作为有效的预测算子。最后，我们给出了基于高斯函数与B-样条的数值算例，以说明并验证理论上的逼近与预测结果。
英文摘要
In this paper, we introduce the Bernoulli--Strang--Fix conditions and their generalized versions for a vector-valued generator $φ=(φ_0,\dots,φ_{ρ-1})$ and a periodic nonuniform sampling set $X$. We use these conditions to establish exact and asymptotic polynomial reproduction properties of sampling Kantorovich operators associated with $(φ,X)$ up to a prescribed degree. We analyze the approximation properties and convergence behavior of these operators in detail. Furthermore, we demonstrate their application to signal prediction from a finite number of past local average samples, showing that sampling Kantorovich operators can also serve as effective prediction operators. Finally, we present numerical examples based on Gaussian functions and B-splines to illustrate and validate the theoretical approximation and prediction results.
Comments20 pages, 3 figures
