Constructing Embedded Lattice-Based Algorithms for Multivariate Function Approximation with a Composite Number of Points

Abstract

We approximate d-variate periodic functions in weighted Korobov spaces with general weight parameters using n function values at lattice points. We do not limit n to be a prime number, as in currently available literature, but allow any number of points, including powers of 2, thus providing the fundamental theory for construction of embedded lattice sequences. Our results are constructive in that we provide a component-by-component algorithm which constructs a suitable generating vector for a given number of points or even a range of numbers of points. It does so without needing to construct the index set on which the functions will be represented. The resulting generating vector can then be used to approximate functions in the underlying weighted Korobov space. We analyse the approximation error in the worst-case setting under both the \(L_2\) and \(L_{\infty }\) norms. Our component-by-component construction under the \(L_2\) norm achieves the best possible rate of convergence for lattice-based algorithms, and the theory can be applied to lattice-based kernel methods and splines. Depending on the value of the smoothness parameter \(\alpha \), we propose two variants of the search criterion in the construction under the \(L_{\infty }\) norm, extending previous results which hold only for product-type weight parameters and prime n. We also provide a theoretical upper bound showing that embedded lattice sequences are essentially as good as lattice rules with a fixed value of n. Under some standard assumptions on the weight parameters, the worst-case error bound is independent of d.