Besov regularity of multivariate non-periodic functions in terms of half-period cosine coefficients and consequences for recovery and numerical integration
Martin Schäfer, Tino Ullrich
TL;DR
The paper develops a Besov-type framework for non-periodic multivariate functions on $[0,1]^d$ using the half-period cosine system, and proves a periodization principle that maps $S^{r}_{p,q}B_{\mathrm{hpc}}([0,1]^d)$ onto the even subspace of the periodic Besov space on $\mathbb{T}^d$, enabling transfer of periodic-domain techniques to the non-periodic domain. It defines and analyzes the HPC Besov spaces, establishes their relationships to classical Besov spaces via embedding and identification results, and demonstrates concrete applications to HPC approximation, weighted least-squares reconstruction, and tent-transformed cubature, including optimal convergence rates for tent-transformed digital nets. A central technical pillar is the Chui-Wang wavelet characterization of multivariate $S^{r}_{p,q}B(\mathbb{R}^d)$, which provides the necessary tools for the CW-based periodization proofs and the associated function-space embeddings. Overall, the work extends periodic Besov methods to non-periodic domains through the HPC framework, with implications for recovery, sampling, and high-dimensional numerical integration. The synthesis of periodization, wavelet characterizations, and HPC embeddings yields a coherent approach to non-periodic function approximation and cubature with provable rates in a broad parameter range.
Abstract
In the setting of $d$-variate periodic functions, often modelled as functions on the torus $\mathbb{T}^d\cong[0,1]^d$, the classical tensorized Fourier system is the system of choice for many applications. Turning to non-periodic functions on $[0,1]^d$ the Fourier system is not as well-suited as exemplified by the Gibbs phenomenon at the boundary. Other systems have therefore been considered for this setting. One example is the half-period cosine system, which occurs naturally as the eigenfunctions of the Laplace operator under homogeneous Neumann boundary conditions. We introduce and analyze associated function spaces, $S^{r}_{p,q}B_{\mathrm{hpc}}([0,1]^d)$, of dominating mixed Besov-type generalizing earlier concepts in this direction. As a main result, we show that there is a natural parameter range, where $S^{r}_{p,q}B_{\mathrm{hpc}}([0,1]^d)$ coincides with the classical Besov space of dominating mixed smoothness $S^{r}_{p,q}B([0,1]^d)$. This finding has direct implications for different functional analytic tasks in $S^{r}_{p,q}B([0,1]^d)$. It allows to systematically transfer methods, originally taylored to the periodic domain, to the non-periodic setup. To illustrate this, we investigate half-period cosine approximation, sampling reconstruction, and tent-transformed cubature. Concerning cubature, for instance, we are able to reproduce the optimal convergence rate $n^{-r}(\log n)^{(d-1)(1-1/q)}$ for tent-transformed digital nets in the range $1\le p,q\le\infty$, $\tfrac{1}{p}<r<2$, where $n$ is the number of samples. In our main proof we rely on Chui-Wang discretization of the dominating mixed Besov space $S^{r}_{p,q}B(\mathbb{R}^d)$, which we provide for the first time for the multivariate domain.