Reproducing kernel Hilbert spaces (RKHSs) are very important function spaces, playing an important role in machine learning, statistics, numerical analysis and pure mathematics. Since Lipschitz and H\"older continuity are important regularity properties, with many applications in interpolation, approximation and optimization problems, in this work we investigate these continuity notion in RKHSs. We provide several sufficient conditions as well as an in depth investigation of reproducing kernels inducing prescribed Lipschitz or H\"older continuity. Apart from new results, we also collect related known results from the literature, making the present work also a convenient reference on this topic.

翻译：再生核希尔伯特空间是极为重要的函数空间，在机器学习、统计学、数值分析和纯数学中发挥着关键作用。鉴于Lipschitz连续性与Hölder连续性作为重要的正则性属性，在插值、逼近和优化问题中具有广泛应用，本文系统研究了再生核希尔伯特空间中这两种连续性概念。我们提出了若干充分条件，并对诱导特定Lipschitz或Hölder连续性的再生核进行了深入探究。除新的研究成果外，本文还汇集了文献中已有的相关结论，使其成为该主题的便捷参考文献。