This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.

翻译：本文对多项式函数型回归进行了全面论述，最终建立了一个新的有限样本界。该边界涵盖了多个方面，包括一般光滑性条件、容量条件以及正则化技术。通过这一工作，本文还扩展并推广了线性函数型回归领域中的若干结论。此外，我们提供了数值证据，表明使用更高阶多项式项有助于提升性能。