Besov regularity of multivariate non-periodic functions in terms of half-period cosine coefficients and consequences for recovery and numerical integration
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| Formato: | Preprint |
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2025
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| _version_ | 1866913777900847104 |
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| author | Schäfer, Martin Ullrich, Tino |
| author_facet | Schäfer, Martin Ullrich, Tino |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_03903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Besov regularity of multivariate non-periodic functions in terms of half-period cosine coefficients and consequences for recovery and numerical integration Schäfer, Martin Ullrich, Tino Numerical Analysis Functional Analysis 41A10, 41A25, 42C10, 42C40, 42B05, 42B35, 46E35, 65D32 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. |
| title | Besov regularity of multivariate non-periodic functions in terms of half-period cosine coefficients and consequences for recovery and numerical integration |
| topic | Numerical Analysis Functional Analysis 41A10, 41A25, 42C10, 42C40, 42B05, 42B35, 46E35, 65D32 |
| url | https://arxiv.org/abs/2504.03903 |