In this article, we consider the reconstruction of a bandlimited function from its finite localized sample data. Truncating the classical Shannon sampling series results in an unsatisfactory convergence rate due to the slow decay of the sinc function. To overcome this drawback, a simple and highly effective method called the Gaussian regularization of the Shannon series was proposed in engineering and has received remarkable attention. It works by multiplying the sinc function in the Shannon series with a regularized Gaussian function. L. Qian established the convergence rate of for this method, where δ<π is the bandwidth and n is the number of sample data. C. Micchelli et al. proposed a different regularized method and obtained the corresponding convergence rate of . This latter rate is by far the best among all regularized methods for the Shannon series. However, their regularized method involves the solving of a linear system and is implicit and more complicated. The main objective of this article is to show that the Gaussian regularization of the Shannon series can also achieve the same best convergence rate as that by C. Micchelli et al. We also show that the Gaussian regularization method can improve the convergence rate for the useful average sampling.

Multiview attributed graph clustering is an important approach to partition multiview data based on the attribute characteristics and adjacent matrices from different views. Some attempts have been made in using graph neural network (GNN), which have achieved promising clustering performance. Despite this, few of them pay attention to the inherent specific information embedded in multiple views. Meanwhile, they are incapable of recovering the latent high-level representation from the low-level ones, greatly limiting the downstream clustering performance. To fill these gaps, a novel dual information enhanced multiview attributed graph clustering (DIAGC) method is proposed in this article. Specifically, the proposed method introduces the specific information reconstruction (SIR) module to disentangle the explorations of the consensus and specific information from multiple views, which enables graph convolutional network (GCN) to capture the more essential low-level representations. Besides, the contrastive learning (CL) module maximizes the agreement between the latent high-level representation and low-level ones and enables the high-level representation to satisfy the desired clustering structure with the help of the self-supervised clustering (SC) module. Extensive experiments on several real-world benchmarks demonstrate the effectiveness of the proposed DIAGC method compared with the state-of-the-art baselines.

Incomplete multiview clustering (IMC) methods have achieved remarkable progress by exploring the complementary information and consensus representation of incomplete multiview data. However, to our best knowledge, none of the existing methods attempts to handle the uncoupled and incomplete data simultaneously, which affects their generalization ability in real-world scenarios. For uncoupled incomplete data, the unclear and partial cross-view correlation introduces the difficulty to explore the complementary information between views, which results in the unpromising clustering performance for the existing multiview clustering methods. Besides, the presence of hyperparameters limits their applications. To fill these gaps, a novel uncoupled IMC (UIMC) method is proposed in this article. Specifically, UIMC develops a joint framework for feature inferring and recoupling. The high-order correlations of all views are explored by performing a tensor singular value decomposition (t-SVD)-based tensor nuclear norm (TNN) on recoupled and inferred self-representation matrices. Moreover, all hyperparameters of the UIMC method are updated in an exploratory manner. Extensive experiments on six widely used real-world datasets have confirmed the superiority of the proposed method in handling the uncoupled incomplete multiview data compared with the state-of-the-art methods.