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：本文对多项式函数回归进行了全面论述，最终建立了一个新颖的有限样本界。该界限涵盖了多个方面，包括一般光滑性条件、容量条件以及正则化技术。在此过程中，它还扩展并推广了线性函数回归背景下的若干发现。我们还提供了数值证据，表明使用高阶多项式项可以带来性能提升。