标题： 非周期多元函数的 Besov 正则性：基于半周期余弦系数以及对重构和数值积分的影响 显示英文标题

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

摘要： 在多元周期函数 $d$-变元的背景下，通常将其建模为环面 $\mathbb{T}^d\cong[0,1]^d$ 上的函数，经典的张量化傅里叶系统是许多应用中的首选系统。转向 $[0,1]^d$ 上的非周期函数时，傅里叶系统不再那么适合，正如边界处出现的吉布斯现象所表明的那样。因此，对于这一背景，已经考虑了其他系统。一个例子是半周期余弦系统，它自然地作为拉普拉斯算子在齐次诺依曼边界条件下的特征函数出现。我们引入并分析了相关的函数空间 $S^{r}_{p,q}B_{\mathrm{hpc}}([0,1]^d)$，该空间推广了支配混合 Besov 型的概念。作为一个主要结果，我们证明了存在一个自然的参数范围，在这个范围内，$S^{r}_{p,q}B_{\mathrm{hpc}}([0,1]^d)$ 与经典的支配混合光滑度 Besov 空间 $S^{r}_{p,q}B([0,1]^d)$ 相同。这一发现对 $S^{r}_{p,q}B([0,1]^d)$ 中的不同泛函分析任务有直接影响。它允许系统地将原本针对周期域的方法转移到非周期设置中。为了说明这一点，我们研究了半周期余弦逼近、采样重构和帐篷变换立方化。 关于数值积分（cubature），例如，我们能够重现最优的收敛率 $n^{-r}(\log n)^{(d-1)(1-1/q)}$，对于在范围 $1\le p,q\le\infty$， $\tfrac{1}{p}<r<2$内的帐篷变换数字网（tent-transformed digital nets），其中 $n$是样本数量。 在我们的主要证明中，我们依赖于 Chui-Wang 对占主导的混合 Besov 空间 $S^{r}_{p,q}B(\mathbb{R}^d)$的离散化处理，这是我们首次为多元域提供这种处理方式。 显示英文摘要

摘要： 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.