Besov regularity of multivariate non-periodic functions in terms of half-period cosine coefficients and consequences for recovery and numerical integration

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, $S_{p,q}^r{B}_{\mathrm{hpc}}([0,1{]}^d)$, 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 $S_{p,q}^r{B}_{\mathrm{hpc}}([0,1{]}^d)$ coincides with the classical Besov space of dominating mixed smoothness $S_{p,q}^{r}B([0,1{]}^d)$. This finding has direct implications for different functional analytic tasks in $S_{p,q}^{r}B([0,1{]}^d)$. 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 $1\le p,q\le \infty$, $\frac{1}{p}<r<2$, where $n$ is the number of samples. In our main proof we rely on Chui-Wang discretization of the dominating mixed Besov space $S_{p,q}^{r}B({\mathbb{R}}^d)$, which we provide for the first time for the multivariate domain.

49 pages