AI 中文总结
本研究将加权半周期余弦空间与加权无锚Sobolev空间的范数等价关系推广至最高为2的分数阶光滑性，建立其与整数阶Sobolev空间的连续桥梁，实现了相关误差界与易处理性结果的转移。
AI 中文摘要
加权半周期余弦空间常被用于拟蒙特卡罗方法理论中，以处理非周期函数的多元积分与逼近问题。对于整数阶光滑性，已有文献证实其范数等价于某些加权无锚Sobolev空间。本研究将该等价关系推广至最高为2的分数阶光滑性，通过引入Slobodeckij型半范数的显式表示，证明了半周期余弦空间与对应加权无锚Sobolev空间之间的范数等价性。该Sobolev范数表示阐明了分数阶正则性如何决定边界约束及（非）周期结构的存在与否。此外，本研究还探究了当光滑性参数趋近于整数边界时这些分数阶空间的极限行为，建立了其与经典整数阶Sobolev空间的连续桥梁。这些等价结果使我们能够将多元积分与函数逼近的近最优误差界及易处理性结果从半周期余弦空间设置转移至新引入的分数阶Sobolev空间。
英文摘要
The weighted half-period cosine space has often been employed in the theory of quasi-Monte Carlo methods for multivariate integration and approximation of non-periodic functions. For integer-order smoothness, its norm equivalence to certain weighted unanchored Sobolev spaces has been established in the literature. In this work, we extend this equivalence to fractional-order smoothness up to $2$. By introducing an explicit representation via Slobodeckij-type seminorms, we prove a norm equivalence between the half-period cosine spaces and the corresponding weighted unanchored Sobolev spaces. Our Sobolev norm representation clarifies how the fractional regularity dictates the presence or absence of boundary constraints and (non-)periodic structures. Furthermore, we investigate the limiting behavior of these fractional spaces as the smoothness parameter approaches integer boundaries, establishing a continuous bridge to the classical integer-order Sobolev spaces. These equivalence results enable us to transfer the near-optimal error bounds and tractability results for multivariate integration and function approximation from the half-period cosine settings to our newly introduced fractional Sobolev spaces.
Comments43 pages, 3 figures
