🤖 AI Summary
This work addresses the problem of high-accuracy signal approximation and prediction under non-uniform periodic sampling by proposing a unified framework based on sampling Kantorovich operators. It extends the classical Bernoulli–Strang–Fix condition for the first time to vector-valued generators in the context of non-uniform sampling and rigorously establishes that the proposed operator possesses both exact and asymptotic polynomial reproduction capabilities. Consequently, the method simultaneously achieves accurate function approximation and signal prediction from local average samples. Theoretical analysis, complemented by numerical experiments using Gaussian kernels and B-splines, demonstrates that the proposed approach exhibits superior performance in both approximation accuracy and prediction capability.
📝 Abstract
In this paper, we introduce the Bernoulli--Strang--Fix conditions and their generalized versions for a vector-valued generator $\varphi=(\varphi_0,\dots,\varphi_{ρ-1})$ 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 $(\varphi,X)$ 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.