In this paper, we describe a novel approach to classical approximation theory of periodic univariate and multivariate functions by trigonometric polynomials. While classical wisdom holds that such approximation is too sensitive to the lack of smoothness of the target functions at isolated points, our constructions show how to overcome this problem. We describe applications to approximation by periodic basis function networks, and indicate further research in the direction of Jacobi expansion and approximation on the Euclidean sphere. While the paper is mainly intended to be Communicated by S.K. Jain.

Our previous multiscale graph basis dictionaries/graph signal transforms-Generalized Haar-Walsh Transform (GHWT); Hierarchical Graph Laplacian Eigen Transform (HGLET); Natural Graph Wavelet Packets (NGWPs); and their relatives-were developed for analyzing data recorded on nodes of a given graph. In this article, we propose their generalization for analyzing data recorded on edges, faces (i.e., triangles), or more generally κ-dimensional simplices of a simplicial complex (e.g., a triangle mesh of a manifold). The key idea is to use the Hodge Laplacians and their variants for hierarchical partitioning of a set of κ-dimensional simplices in a given simplicial complex, and then build localized basis functions on these partitioned subsets. We demonstrate their usefulness for data representation on both illustrative synthetic examples and real-world simplicial complexes generated from a co-authorship/citation dataset and an ocean current/flow dataset.